The Team Fortress 2 Beta game is where the devs test changes to the game prior to rolling them out officially. It isn't intended to replace the "real" game for normal play; the rules for acquiring items are very different, for example, to facilitate testing.
More detailed information is available from the official wiki.
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.
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Best Answer
Functionally, there is no difference between the two. I believe buying something from the Mann Co. Store makes said item uncraftable for a week, but that might have changed in my absence from TF2. And as for the price difference, I'm fairly certain it follows the general trend that uncraftables are cheaper than regular items.