Since you got Bahamut, I'll expand upon what the Rare Item ability does, because it's not immediate.
So, each enemy has a % chance to drop something after being defeated if they're not Mugged during the fight. What can be dropped depends on the enemy and on the level of the enemy. The drops are organized in 4 (slots 0, 1, 2, 3 internally) slots, each with a fixed % chance of being selected: Slot 0 has a 69.53% (178/256) chance, Slot 1 is 19.92% (51/256), Slot 2 is 5.85% (15/256) and Slot 3 is 4.68% (12/256). This is the distribution without Rare Items in the mix. Often the rare items are in slot 2-3, and sometimes the only difference is that higher slots have more of an item to drop.
What Rare Items does and why the name is misleading is altering the chances for every slot, setting them to: Slot 0 to 50% (128/256), Slot 1 to 44.53% (114/256), Slot 2 to 5.46% (14/256) and Slot 3 to 0% (0/256).
As example, let's use a enemy you know well: the Malboro.
Level 30+ Malboros drop items 99% of the time when defeated, and the slots are organized like this:
Drop Table, normal | |
---|
Slot 0 (69.53%) | Malboro Tentacle x8 |
Slot 1 (19.92%) | Curse Spike x10 |
Slot 2 (5.85%) | Hypnocrown x1 |
Slot 3 (4.68%) | Hypnocrown x2 |
As you can see, it's very common to get Malboro Tenteacles when kiling a Malboro, while Curse Spikes are not very frequent and the Hypnocrown is rare, and sometimes you do get 2 of them.
With Rare Items equipped the drop table changes to this:
Drop Table, with Rare Items | |
---|
Slot 0 (50%) | Malboro Tentacle x8 |
Slot 1 (44.53%) | Curse Spike x10 |
Slot 2 (5.46%) | Hypnocrown x1 |
Slot 3 (0%) | Hypnocrown x2 |
Basically you get a more than double chance for Curse Spikes, in exchange for a lower rate of Malboro Tenctacles and a halved rate for Hypnocrowns; also you'll neve get 2 Hypnocrowns.
What come out is that the ability works well if you need "rare" (slot 1) drops from enemies, at the cost of the actual rare drops in slot 2 and 3.
Now, sometimes this can be useful,
@Egleris has posted some examples before, and from the example here Malboros become a somewhat realiable way to farm Curse Spikes, though not the best.
Mugging works off similar mechanics: when you Mug an enemy you have a % chance to get an item, and if you get an item the game chooses from one of 4 slots with the same weights as the drop table. Each enemy has its own Mug slots, which can be the same items dropped or could be different.
To use the Malboro (Mug chance 25%) again as example:
Mug Table, normal | |
---|
Slot 0 (69.53%) | Malboro Tentacle x2 |
Slot 1 (19.92%) | Malboro Tentacle x2 |
Slot 2 (5.85%) | Malboro Tentacle x4 |
Slot 3 (4.68%) | Malboro Tentacle x4 |
In practice if you Mug a Malboro you have a 89.45% for 2 Malboro Tentacles and a 10.54% for 4, since Slot 0 and 1 have the same items and slot 2 and 3 do too.
Rare Items does also alter the Mug tables, with the same altered chances. So for Malboros the Mug table becomes:
Mug Table, with Rare Items | |
---|
Slot 0 (50%) | Malboro Tentacle x2 |
Slot 1 (44.53%) | Malboro Tentacle x2 |
Slot 2 (5.46%) | Malboro Tentacle x4 |
Slot 3 (0%) | Malboro Tentacle x4 |
It comes out to a 94.53% for 2 Malboro Tentacles and a 5.46% for 4.
In general, due to how Mug slots are set, Rare Items has less on an impact on Mug than it does on drops: a lot of enemis have the same item in the whole Mug table, only differing in quantitiy, and some have just the same exact entry for the whole Mug table.
The biggest case where you don't want Rare Items is if you want to Mug Blitzes for a Power Generator, since it's a Slot 3 Mug and Rare Items makes so yu'll never see it. You shouldn't care about this case, you already got one from the Lunatic Pandora and it refines in items you can get more easily in other ways.
This came out a lot longer than I tought. Still the ability is misleading and it touches mechanics that aren't explained in the slightest, so it's needed. Mug works among smilar lines, so at this point an explanation was in order. And hey, Bahamut has Mug too.