Well with Amethyst you can summon Atomos who can do great damage but its based on HP and won't be very useful against certain bosses who are immune. With opals I can summon Shiva who does decent damage, but not enough for me to consider. Sapphires let you summon Fenrir who does ok earth damage, but is useless against flyers. Aquamarine nets you a Leviathan who does pretty good damage but the MP cost is great and I don't think the trade is fair. He has a pretty decent animation. Peridot and Ramuh well I found him to be the weakest of them, but his introduction is cool. Emerald I think gave Carbuncle a different attack I can't remember what was different, but I never found much use for the summon in general. Garnets let you summon a Bahamut and he does great damage and its non-elemental so you don't have to worry about it being absorbed by a specific enemy.