In fact, there are 3 types of movement orders in the game:

attacking - units have a certain degree of freedom when maneuvering and will engage enemy units on contact;
defensive - units are more restrained during the maneuver and are less distracted by enemy firee, upon arrival at the destination they will take up positions in any available cover or suitable terrain features (craters, trenches, buildings etc);
movement - units move directly from point A to point B.

All of the default orders differ ONLY by use of active modifiers, the exception is the "defense" command, where units take cover in defensive positions upon completion. Explanations for orders and modifiers can be viewed in the pop-up menu by hovering over them with the mouse (see also manual page 21-26).

What are the differences between the various attack orders:

"Assault" - attacka the target directly without significant maneuvering while maintaining a tight line formation.
"Attack" - attacks the target with more freedom to maneuver, loosley maintains a line formation and uses smoke when available.

More detailed:
Assault is essential in those cases where you need your units to quickly advance on the enemy position without unnecessary maneuvering. For example, you can use the Assault command for heavy tanks, in cases where there is no credible threat of them running into heavy AT weapons. Being fairly immoblie, it pays to keep maneuvering to a minimum, this will encourage them to keep their frontal armour oriented towards the target area. So it makes sense to use the assault order for attacks over relatively short distances, in other words to storm the enemy positions in a dense powerful and fast moving formation, if used correctly it can be overwhelming against static targets and allow for quick establishment of numerical superiority. The assault command can still be very effective when used over greater distances, where the range of weapons can be fully exploited, especially in situations where the enemy has no effective AT weaopns and the terrain allows for movement with less maneuvering, for example across open fields and plains.

Note: It is Important to understand that when the 'maneuver' modifier is disabled, units will try much harder to maintain a rigid formation and they will make individual maneuvers much less often, so be careful about sending them across challenging terrain, their attempts to maintain formation and facing towards the enemy can make it more likely that they will fall victim to obstacles like tank traps, craters and trenches.

Attack order is best employed where the distance to the enemy is large and the liklihood of fire from the flanks is high, as your units will be free to engage enemy umits, maneuver to face threats and use smoke to shiled the advance, however it can and usually will result in the exposing of your units sides or rear to enemy fire. The use of this command makes the most sense for example for infantry at medium to long range (when the enemy possesses significant weight of small arms fire, heavy weapons and/or artillery), or when you attack with tanks against heavy anti-tank weapons which you expect to be a threat from any number of positions and directions, and it doesn't really matter exactly how your units are oriented for the duration of the advance. In such situations maneuvering can only be beneficial, and will reduce thst likelihood of your units being sitting ducks to attacks from the sides or even rear.

Note: if you attack with AFV's across perticularly uneven or obstacle littered terrain, the "Attack" order will generally be more optimal than "Assault", because units will attempt to go around obstacles, but this will not guarantee that no one will get stuck or will not turn over, so always try to give troops the most suitable route and always use roads where possible.

Correction of orders
Sometimes it makes sense to tailor the default orders by changing the modifiers.
For example, when attacking through a very narrow section of terrain like a ravine or marching via road, you might need to select "column" formation for that secton of the route, and then when the terrain allows change the formation modifier back to "line" for the next command. When attacking it can also make sense to change the number of 'rows' for AFVs, this gives you the opportunity to form a wedge (formation line - number of rows 2, or 3 for a deeper wedge).
Formation "without formation" means that units will move to the target without worrying about maintaining their shape, which can make movement more fluid.
With "assault" commands in some cases it can make sense to enable the 'Use smoke' modifiers, but you should to take the direction and strength of the wind into account, if your units are attaing against or with the wind smoke screens can become useless or even harmful.

Retreating under fire
When using the 'use smoke' modifier with a "defense" order, the unit will pop smoke upon completion of the command, and ideally this could be used to cover a withdrawal. For it to work best it is necessary to chain two orders on the vector of retreat, one short ''defense'' command with smoke enabled to where you want the smoke deployed, and a second order with smoke disabled and 'fast move' modifier enabled ending in the chosen place of safety. In this case the unit would move a short distance, deploy the smoke screen, then rapidly continue back through the smoke towards their eventual defensive positions. A panic retreat can also be made using the ''attack'' order with a fast movement modifier, although your units can get too embroiled in the fight. A better option would be to combine these orders for a large detachment, so that one branch retreats with an "attack" order thereby shielding other troops smoke and gunfire, while the other units withdraw using "defense" or "movement" commands.
Don't forget that you can change the density and type of formation for "defense" orders, to make awide/narrow deep/shallow defensive front.

IMPORTANT
If you issue the order using the "space" + LMB or hotkeys, you can set the direction/vector of the front for this order, holding the RMB and pulling in the appropriate direction.

Orders on the move - Let's explane 2 of 5.

"Covert movement" - With this command, the squad will attempt to move through ravines, forests, bushes, take cover in the folds of the terrain. An additional modifier that makes sense to enable is the "fast move" to accelerate the movement.

Note: it is IMPORTANT to understand that if there is no cover (ilke those features listed above) en route to the destination, or the points are located close to each other, then giving this order makes no sense, it will function only as a simple move order, with any change in formation and density dictated by the given modifiers.

"Recon" - Advance until contact with the enemy. Here, by the way, the default order is not quite optimal (in the line of average density), when contact with the enemy under fire gets the whole group. So it makes sense to change order modificators to the column + wide formation, then the units move in a column is exhausted, when contact with the enemy under the fire reaches the head of the column. Sometimes it makes sense to add "fast motion" and/or "stealth instant order" modifiers.

An example of implementation of reconnaissance tasks: head watch is the order of intelligence (column), followed by 150-300m is a group of cover in the assault/movement line. Therefore, when contact with the enemy head watch will cover fire team.

Note: to order intelligence does not make sense if units can see the enemy (they will suspend the execution of the modifier).

IMPORTANTLY!!!

• You have to understand that giving different types of orders you specify different formations types (by default), therefore units should not just get from point A to point B, but also to change the formation. So be careful, and correctly combine orders, otherwise the line can turn into a column, and a wide construction in a dense pile, and all this can happen in the middle of the battlefield.
  
• An important feature is that when driving in a column in the absence of subsequent orders, the endpoint of the path units can temporarily fairly tightly grouped in one place, and become a good target. Therefore, if there is danger of meeting with the enemy, always give the extra order of "defense" at the last point.
  
• So it is necessary to understand that moving over rough terrain units try to stick to a formation, and if the terrain is impassable for the vehicles, it may get stuck or roll over, including well pick a route for traffic, and use the "COLUMN" formation for units so it is easier to avoid obstacles, or choose "FREE FJRMATION", then the vehicles will not try to take the information and will be able to maneuver freely.

All modifiers and their combinations are described in detail in the manual for the game (p. 23-26)

Let's focus only on some of them:

The Modifier "Quickly"
When you turn on this modifier at night, the vehiclest turns on the lights, because with fast movement without light it can get into an accident.

The Modifier "Maneuvering"
When this modifier is activated, the units are free to maneuver with respect to a given motion vector and are able to bypass obstacles better.
However, remember that in combat the excessive manoeuvring is fraught with the fact that the vehicle can substitute under enemy fire less protected projection.

The modifier "Pop smoke"
If you enable this modifier units put a smokescreen on the way, this is useful in case of strong fire resistance when attacking or retreating. However, this is not useful if you have superiority and the enemy can have on you powerful fire exposure (due to low number or lack of serious firepower), thus you only worsen a review of his troops and reduce the intensity of the fire.

Instant order " Covertly"

This modifier forces the infantry to lie down, and the technique to close the hatches. The combination of the modifier "COVERTLY", and the command to move with the modifier" QUICKLY " gives the fact that the infantry lies and moves defections, not only under fire but in any situation. If you combine the modifier "covertly" with the command movement without mod. quickly, the soldiers will move only by crawling.
This is an important point that must be properly combined with the rest of the teams on the move.

If crew of the weapon / tank in which ammunition there are smoke shells to include the modifier "covertly", and then through the commander to give the order "Fire at the district", fire will be conducted smoke shells (but if they aren't unit, won't shoot).

Instant order "Transfer control to the AI fire"

This order prohibits the unit to choose targets and to have its own sighting device, the data about the enemy he only receives from the commander.
This order is effective if the unit is in a closed position (or does not see) the enemy and receives the target designation from the commander who is on the observation post. Fire on the HQ usually conducted by area and the most effective artillery.

If you enable this, you can order through the commander to give the order "Bombard the area" virtually to any units, including infantry, tanks, etc.

HOWEVER, this order is harmful in those cases where units can see the target and can fire, because when "Transfer control to the AI fire" units will have their own data for the shooting, and only less precise data of the commander (if the commander does not see the purpose, crew will did not open fire, even when he sees enemy).
Simple example if the enemy came out on a direct tip to your howitzers disconnect "Transfer control to the AI fire" otherwise howitzers will shoot very accurately and with a delay according to the commander of the battery.

"Priority target"

It is also worth mentioning the order "Priority target" forcing units to actively attack the selected troops. To give that order click on the bottom panel of quick orders and hold the LMB encircle enemies by frame.

Units in the game have a developed artificial intelligence, able to make their own decisions, the game does not tolerate to "mous clicking", and the orders given in the game very strongly affect the actions and behavior of the troops. Incorrectly given command will cause inadequate reaction on the battlefield that can lead to unexpected consequences, at the same time correctly given order is capable to change cardinally a situation.

Do not try to fight with the AI of your troops, but understand how it works and learn to interact with it correctly, the result will pleasantly surprise you.