Strength of Claim
An allegation is a weaker claim than an accusation. It’s a statement that someone did something wrong, but there’s no guarantee it’s true. An accusation is a stronger claim, implying a belief that the wrongdoing is true.
Evidence
Allegations don’t necessarily require evidence, while accusations often do. Someone might make an allegation based on suspicion or hearsay, but an accusation usually comes with some kind of proof.
Legality
Allegations can be about anything, not just legal matters. Accusations, however, often refer to criminal activity. For instance, you could allege someone ate your lunch, but you wouldn’t accuse them of that (unless maybe it was a very expensive lunch!).