What is the difference?

I know that with single you can claim a keep. What else is different?

If a player joins single alliance guild and plays alt in another faction, he can still see the guild/chat/store?

Is there something you can only achieve with cross alliance guild?

Every guild can only belong to one faction as far as Cyrodiil goes. The fact that you can have players of other factions in your guild doesn't change that, it can only still be aligned with that one faction.

So basically there is absolutely no use for cross alliance guild? You can do all the same with single alliance + with single you can also be efficient in AvA?

Well i mean if you have friends or make new friends on other factions is nice i guess to have in the same guild to chat and keep in touch as well as access to the guild store for trade. But yeah when a guild is created or some time after that it is set to one faction.

You are in a single alliance guild and you make an alt to another alliance. You can still play in multiple alliances and do trade from another alliance. Right?

So why would anyone do cross alliance guild?