Update: Terraria 1.3 brings with it seriously buffed antlion mandible drops, as well as a whole host of new kinds of antlions to drop them.

Getting enough for whatever you need should now be a lot easier, after running through the Underground Desert a bit.

Kill antlions.

While that answer seems facile, I'm afraid it's not just the easiest way, it's the only way*. And it takes approximately forever.

To speed up the process, you can engage in the construction of a mega project: Start a new world and convert it's entire surface to a desert, seriously increasing antlion spawning. The sand you need can be gathered from the antlions themselves as they start to spawn, as they are capable of producing an unlimited amount of sand. (Although getting any reasonable quantities of sand that way also takes, you guessed it, forever.

*: I say it's the only way, but in truth there is another: Find a server and trade for the mandibles with someone that has already done all of this.

There are 3 main differences IIRC.

In the pre-hardmode, Crimson worlds have Crimson Armor, Deathbringer Pickaxe, Fleshgrinder Hammer, The Meat Ball Flail, and the Crimson Rod. Corruption Worlds have Shadow Armor, Nightmare Pickaxe, The Breaker Hammer, The Ball O' Hurt Flail, and Vilethorn. In Hardmode, Crimson worlds get access to Ichor, where Corruption worlds get access to Cursed Flame. In post-Plantera Hardmode, the Crimson gets Vampire Knives while the Corruption gets the Scourge of the Corrupter.

I expect that the reason you get different answers is because the answer is different at different points during the game. It might also vary depending on your play style and whether you are playing multiplayer or single player.

Let's break it down:

Pre Hard Mode

Armor

+---------------+-------+-------------------------+-------------------------------+
|     Name      | Armor |         Bonus1          |            Bonus2             |
+---------------+-------+-------------------------+-------------------------------+
| Shadow Armor  |    19 | +21% Melee Attack Speed | +15% Movement Speed           |
| Crimson Armor |    19 | +6% Damage              | Greatly Increase Health Regen |
+---------------+-------+-------------------------+-------------------------------+

The attack speed will likely result in higher DPS than the damage increase if you are melee. If you are ranged, Crimson armor is better. Movement Speed is quite nice before you have wings, but the Health Regen obviously wins out here. Comparing those together, I think the Shadow Armor is slightly better for melees (DPS is really important), but it depends on your playstyle.

Pickaxes

+----------------------+---------------+--------+--------------+
|         Name         | Pickaxe Power | Damage | Mining Speed |
+----------------------+---------------+--------+--------------+
| Nightmare Pickaxe    | 65%           |      9 |           15 |
| Deathbringer Pickaxe | 70%           |     12 |           14 |
+----------------------+---------------+--------+--------------+

The Crimson pickaxe is slightly slower, but requires one less hit for higher end ores. This means that for high end ores, it ends up being better, but it is slightly slower for low end blocks. The difference between these is very negligible.

Hammers

The difference for hammers is likewise very small.

Flails

The Meatball is 1 more damage than the Ball o' Hurt (15 vs 16), so there is not a lot of difference here.

Spells

Both spells cost the same amount of mana. The crimson Rod has slightly more damage, but I think Vilethorn is the clear winner here because it can attack through walls. That is a pretty unique trait in Pre-Hard Mode.

Hard Mode

Ichor vs Cursed Flame

There are many things that you can make from these ingredients, but I am going to group them together since they all serve to apply the ichor and cursed flame debuff. The ichor debuff lowers the armor of enemies by 20. Cursed flame applies a damage-over-time effect that stacks with "On Fire" and cannot be put out.

In general, the ichor debuff is much stronger, although it depends on your weapon fire rate. Armor in Terraria works very simply. Evert 2 points of armor reduces the damage from a hit by 1. The ichor debuff makes every attack hit for 10 more damage. This is a large increase in DPS for high fire rate weapons (Minishark and Megashark for example), but is less effective for low rate weapons. Still, though, the Cursed Inferno Debuff is 4 DPS, so you would need to hit less than once every 2 seconds for Cursed Inferno to be better than Ichor.

For melee, ichor is quite good even through the end of the game, since it can be applied by using any melee weapon by using a flask of ichor. For rangers, the high end bullets tend to outclass the ichor bullets.

In multiplayer setups, it is useful to have the melee player use a flask of ichor while having the ranged player use a higher damage bullet like Crystal or Chlorophyte.

Dungeon Items

+--------------------------+--------+---------------+-----------+-----+---------------------------------------+
|           Name           | Damage | # Projectiles | knockback | DPS |           Secondary Effect            |
+--------------------------+--------+---------------+-----------+-----+---------------------------------------+
| Vampire Knives           |     29 | 4-8           | 2.75      | 653 | Heals player for 7.5% of damage dealt |
| Scourge of the Corrupter |     64 | 2-4           | 5         | 505 | Secondary Projectiles are homing      |
+--------------------------+--------+---------------+-----------+-----+---------------------------------------+

Both of these are melee weapons, despite having projectiles. While the scourge of the Corrupter is nice in that it has homing capabilities, the life steal from the vampire knives is so incredibly strong that many, many pages of text have been written arguing about it. Because of the lifesteal from the item, melee players are able to itemize much more offensively. Typically, it is used in combination with a higher DPS weapon like the Terra Blade or Flailron, where you switch to the Vampire Knives when low on life. The scourge of the corrupter becomes obsolete once you have the Flailron (accessible on both worlds) because the Flailron has homing projectiles and much higher DPS.

Conclusion

For casters and rangers, I don't think there is a huge difference between the worlds because you will probably be using other items that are available on both worlds. For Melee users, the crimson is far and away better because the Vampire Knives are incredibly strong and ichor is a much larger DPS increase than cursed flame.