What's the difference between Critical Hits and Mini-Crits?

Critical hits pack more punch, basically.

A critical hit deals 300% the base damage of a weapon, disregarding distance between you and the target and random damage spread. In other words, you'll always deal the same amount of damage no matter your luck and distance, so long as your shot hits your target. Splash damage works normally, except obviously it'll deal thrice the punch.

Unless random crits are disabled, every time you fire your gun* you have a chance to shoot a random critical shot. Chances range from 2% to 15% based on how much damage you've dealt (or your buildings have dealt, or your medigun patients have dealt) in the last 20 seconds. (For melee weapons, the chances are 15% to 65%. Watch out for those wrenches and medic saws!)

_{*For the miniguns, the flamethrowers, the syringe guns, the pistols and other such weapons you only roll for crits once per second. A successful roll, however, grants you two seconds of guaranteed random critical hits.}

Additionally, some weapons have guaranteed ways to obtain critical hits (like all spy knives when attacking from behind), some weapons have no ways to obtain critical hits whatsoever (like the Cow Mangler), and some are in between (the flare gun will crit enemies on fire as well as deal random critical hits.)

A mini-crit is, instead, a simple 35% damage bonus after distance ramp-up (the damage bonus you get for hitting enemies up close) but without distance fall-off (the damage penalty you get for hitting enemies from a distance). Damage spread is also applied.

You can only deal minicrits if something causes you to; the TF2 wiki has a list of such situations. Most notable are: jarate, buff banner, airblasted non-crit projectiles, death marks.

Read more about damage ramp-up and fall-off here.

Taken directly from the official wiki:

Critical hits deal 300% of the base damage of the weapon and ignore both the distance modification and the random damage modification normal shots are susceptible to, though explosive weapon crits still suffer from splash damage fall-off.

Mini-crits add an extra 35% damage to a weapon's damage. Like normal critical hits, they ignore damage fall off, but unlike critical hits, they are still given ramp-up at close range and random spread. Full crits override mini-crits

So, the first difference is in the damage dealt. There's also a difference in the chance of a crit or a mini crit happening:

• Mini-crits are an effect of various weapons and items. More specifically a mini-crit will happen when:

• Hitting Jarate-coated enemies.
• Melee hits midway through your Chargin' Targe or Splendid Screen charge.
• Attacking an enemy while under the effects of a friendly Soldier's Buff Banner.
• Hitting an airborne enemy with the Direct Hit who was propelled airborne by an explosion.
• Attacking an enemy as a Scout while under the effects of Crit-a-Cola.
• Attacking an enemy Scout who is under the effects of Crit-a-Cola.
• Attacking an enemy as a Heavy while under the effects of the Buffalo Steak Sandvich.
• Attacking an enemy Heavy who is under the effects of the Buffalo Steak Sandvich.
• Hitting an enemy as a Pyro with a projectile, other than a stickybomb, reflected with the compression blast.
• Attacking an enemy marked by the Fan O'War.
• Attacking as a Scout while under the hype effect of the Soda Popper.
• Shooting an airborne enemy with the Reserve Shooter during the 3-second mini-crit period it has after switching to it.
• Hitting a burning enemy with the Detonator.
• Shooting enemies with the Cow Mangler 5000's charged shot.
• Attacking a Heavy who is wearing the Gloves of Running Urgently, and within three seconds after he switches weapons.
• Hitting an enemy from long-range with the Flying Guillotine.

Critical hits instead have a base percentage to happen that is increased by the damage dealt in the previous 20 seconds:

Standard weapons have a 2% base chance to crit, while melee weapons have a 15% chance.[1] An additional bonus percentage is added based on the total damage done by the player in the last 20 seconds, which scales from 0% at 0 damage to 10% at 800 damage. For example: dealing 200 damage in the past 20 seconds, the additional bonus percentage is 2.5%. The critical hit chance for melee weapons scales more sharply than that for ranged weapons. Melee critical hit chances can reach up to 65%, meaning the bonus they receive scales from 0% to 50%. Among those weapons that trigger randomly, critical hit probability is determined differently based on the type of weapon. Single-shot weapons, which, for example, include the Shotgun, Revolver, and Rocket Launcher, determine the chance to crit with every shot. Thus, every shot of the Shotgun will have a 2% chance to launch a flurry of critical pellets, assuming no previous damage done. Rapid-fire weapons, however, which include the Minigun, Syringe Gun, Flamethrower, and Pistol check for critical hits once a second. With weapons like these, a positive crit determination results in all shots for the next two seconds being critical.

There are also special cases, that are generally specific to the weapon wielded, so you can find them in the weapons' descriptions.