rarity

I have a question about rarity. Is there any difference between different rarity, like is blue always better than green for the same item lvl?

There's usually no difference between rarities of the same item level. Typically, rarer stuff is higher item level but that is not always the case.

Item level is all that really matters, and items with a colored background typically have a higher item level than ones with gray backgrounds. The colors are basically as follows:

Green: Quest rewards and dungeon drops (including items crafted using items that only drop in dungeons). Some of these can be traded between players. All can be turned in to the grand companies in exchange for seals.

Pink: Aetherial items obtained from dungeons and levequests. These are the same as the item they're based on, with a few additional random stats and often a lower required level to equip. These cannot be traded to other players. These can be traded to the grand company for seals.

Blue: Items of exceptional uniqueness. Artifact armor, darklight armor, and primal weapons are the main places where'd you see this. These cannot be discarded, removed, or sold in any way. Collect them all!

Purple: This color is only used for relic weapons, and I believe for Allagan weapons.

Also, something to note: If an item is crafted, it must be high-quality to compare to non-crafted items of the same item level. Also, for things you can add materia to, the HQ item without materia is what will be on-par with non-crafted items of the same level. Adding materia can make them better (Though, item level does determine the stat caps, so you won't be able to push individual stats past what the non-materiad items have)