Not such an easy question to answer.
The correct term has always been "all right" but the language has evolved and, these days, "alright" would not be considered incorrect by most people. You'll even see it used frequently in popular fiction.
Even so, my spell checker HATES it.
Alright is considered informal, and is considered non-standard English. It is recommended avoiding it.
Both are pretty the same today. It called development and mixture!! I assume it could be matter of formality and informality now!
alright.
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