Nearly no difference at all, in fact alright is suspected to be an incorrect contraction of all right that has just been absorbed into modern usage. Most of the time they can be used interchangeably where alright is used in conversation and informal writing.

I don't think there's any difference. In UK English "That's all right" is commonly used to mean, as Michael says 'that's fine', 'that's good', 'there's no problem'. "That's alright" is just a shortened form of writing the same thing.

I think that "That's all right" = Everything is correct. "That's alright" = It's fine. But I might be wrong..,