This started off in the comments, but I'll spin it off to a separate answer. The other answer well addresses some practical RL-based uses for a Mount, and all of those are true.
Here, I will lay out the actual mechanical uses of a mount according to D&D 5E.
A Standard Mount is a creature like a Riding Horse, Elk, Warhorse, Pony, Mastiff (if you're Small), or other such creature. They have a few basic mechanical benefits.
- Greater carrying capacity. They are a Size larger than you, and thus have a greater carry capacity than you do. This only really matters if you are playing with Encumbrance Rules
- Greater Speed. Riding Horses and Warhorses have a Speed of 60'...double the average speed of a humanoid. Only a high level Monk can keep pace with a horse.
- An Extra (limited) Action. On your Turn, you can order your Mount to Dodge, Disengage, or Dash that can be used for free on your turn. This means you could have your Mount Dash (up to 120' movement speed) and still Attack on your turn. This is a level of mobility that nothing else in the game enjoys. This is a boost to Action Economy, which is one of the most useful things to gain in all of 5E.
Now, what can you do with this?...let's look at a few examples.
- Spellcasting on the move: Spellcasters generally do not want enemies to get close to them. If you are on a Mount and someone gets close to you, your Mount can Disengage, move its speed (usually greater than the speed of whatever is attacking you), and you can still cast a spell that round. And very few enemies are fast enough to keep up with a Mount. This allows a spellcaster to effectively kite enemies that do not have a Ranged option, keeping them at distance while bombarding them with spells.
- Blitz. Enemies with a focus on a Ranged Attack, or enemies who are Spellcasters will try to keep you at range, generally because they are very vulnerable in melee. Goblins are notorious for this...their skill-set makes them fantastic skirmishers who will kite superior opponents until they are dead. If you are mounted, they simply can't move fast enough to stay away from you. You can close with a target far faster than it can get away.
- The Lance. The effectiveness of this weapon, paired with a Mount, is simply spectacular. It is a weapon that can be wielded 1-handed while mounted, has Reach 10', and deals 2d6 damage. The most important part of that feature is the Reach of the weapon. A benefit of a Reach weapon is that it allows you to strike at an enemy from outside of their Threatened space. When you couple this with the extreme mobility of a Mount, and you have an unparalleled ability for striking at enemies without exposing yourself to a counter attack. You command your mount to Dash and ride towards the enemy, always staying at least 10' away from them. This allows you to strike at an enemy as you pass them by, while never getting close enough to risk an Opportunity Attack, and ending your movement so far away from them that they could not possibly make it to you on their next turn. If you have the space to maneuver, melee-focused enemies are trivialized by this tactic.
And all of this is possible without the Mounted Combat Feat. Though if you intend to fight from a mount often, that feat is quite fantastic.
Do note that mounts tend to have rather poor Armor Class...but Barding is only 4x the cost of a standard suit of armor. You can take a Warhorse from AC 11 to AC 16 for only 300 gp...which is chump change to an Adventurer as you get higher level.
A Paladin's 'Find Steed' mount deserves special consideration for four reasons.
- You can 'instinctively coordinate' with your mount
- Your mount is effectively immortal (if killed...you've lost it for one day at the most, when you can prepare Find Steed again and summon it back)
- Your Mount has an Int of 6.
- It is perfectly loyal to you.
The rules for Mounted Combat specify that an intelligent mount can operate 'independently' while being ridden. This means that it has its own position in the Initiative Order and acts with its full battery of Actions and options for acting. The normal risk of working with an intelligent mount is that it may not do what you want it to do...and you're stuck on its back. It may run off, it may chase something you don't want it to, and so on.
With a Paladin's mount, this is not an issue. It is perfectly loyal, and it 'instinctively coordinates' with you. This allows you to use your mount the way a Knight would use their mount in the Medieval era. Here is an example of how this might play out, assuming you are armed with a Lance and riding a Warhorse.
You come up earlier in the Initiative Order than your mount (if not, command your mount to defer its turn until right after yours), and Ready an Action to Attack a specific enemy when it comes into range. Then you end your turn and let your Mount move. You instruct your Mount to charge and trample the same enemy you Readied an Action to attack. It charges and, at 10' out, you lash out with your Readied Action and Attack your target...possibly hitting your target with some manner of Smite while you're at it. Then your Warhorse reaches them and attacks with its Hooves. If it hits, the target makes a Save to not be knocked prone. If it fails, your horse makes a second attack against it with its Hooves, then continues on its way.
This is not what you'll want to do every time...sometimes you don't want to try to trample a foe, you want to stick to the Hit and Ride tactics normally used with a Standard Mount...but it does give you extra options not available to anyone else.
Bear in mind, having your Mount actually trample someone means it won't be staying out of attack range, and won't have its Action free to Disengage, meaning you are subjecting yourself to a possible opportunity attack. But, this is a risk you can afford to take, especially if your Mount is armored...because if your Mount is slain, you can just summon it back in the morning.
For extra goodness, pick up the Mounted Combatant Feat and enjoy Advantage on every Attack roll on anything smaller than your Mount (which, if you're riding a Large mount...is the majority of enemies), and the ability to force attackers to aim at you (and your much larger pool of hitpoints) rather than at your Mount.