There are many strategies that could make use of them. most were covered by Mr.J so i won't repeat them here, just upvote his answer for the information he provided.
- Very Strong
- big curved pointy things on the front
- Reasonably quick for its size
- Comfortable in colder climates where other mounts require more care
- Quite thick skinned
- Needs a capable handler
- easily outpaced by other smaller mounts
- Needs to constantly eat (if anything like real life elephants)
These are very basic pros and cons. and I'm basing them mostly on modern elephants
When trying to figure out how effective a strategy or combat unit would be, I find it easier to think of how easily you could defend against them. in this case i'm looking at the mammoths themselves, not the strategies that they are used in. and going from the perspective of an experienced multi-planetary expeditionary force. not a band of thugs.
A huge angry beast has been sent charging towards us, how do we defend against it. well similar to regular cavalry, get out of its way, a beast that big and heavy cannot stop and turn quickly, so let it charge past us, and it will probably keep going. and if we can get a few spears and arrows into it as it passes means it will weaken as each soldier it passes gets a shot into it. yeah we'll lose a few that get caught by the tusks and trampled but for the most part it shouldn't cause too much of an issue and we'll reform the line fairly quickly.
Obviously if the enemy troops were running directly behind the mammoths then that would mean they could get behind our front line which if that is pike men would mean the couldn't turn easily and cause a fair bit of damage
They could work as a battering ram fairly well and drag the seige towers into place but... as others have suggested, Elephants have fairly sensitive feet. so simple spike traps and caltrops would cause huge issues for them
Once in range of defense catapults or balistas then they are a large target to hit and unlikely to take the damage injured, and an injured mammoth is more likely to turn and ran trampling their own troops so they would be target number one for our bigger weapons.
But also once they got close to the walls then use fire... lots and lots of fire, Mammoths are hairy beast, the hair would catch alight and it would panic and run again damaging the enemy troops.
Armoured Elevated Archer Platform
Its hanging back and its archers are taking shots at us, we try and hit them and hit the mammoth, which is armoured and the shots aren't inflicted much damage, but to be fair, its only a couple of archers up there and they are fairing from a moving platform, so they are not as accurate, so just keep wanging arrows at it and eventually we'll get the archers, and providing we are not a densely packed force then the elevated archers will not be that effective
This would be fairly similar to both the charging and elevated platforms, but they wouldn't be that effective here either. we have horses. which are more agile and faster, we can literally run rings around the mammoths while the riders stab and shoot the driver, or go for the eyes of the mammoth. again, if we can panic the beast (not easy with a well trained beast but still possible) then they'll likely do as much damage to the enemy as to us.
Theoretically capable of dragging very heavy loads compared to horses, means the enemy can transport huge amounts of goods and equipment, apart from laying traps there is nothing we can do to stop them, even the traps would probably be found by the soldiers in front.
If again we assume they are similar to Elephants in terms of diet, then this will be the biggest drawback to them as an effective mount in a lot of environments. so the only decent defense would be to strip the land or burn in ahead of the invading force so that the mammoths don't have enough food to eat.
They would be most useful as beast of burden as opposed to actual combat units