I suppose in theory that there are typing errors you could make that wouldn't be typos. Typing past the right edge of the paper (on a typewriter) would be a kind of typing error, but we wouldn't call it a typo. But for at least 99% of cases, they are the same thing.
Typo is a clipping of typographical: in other words, it is associated with typography (arranging the type), not necessarily with typing. I agree, however, that these days, there isn't a lot of difference between them.
I consider them interchangeable in meaning, but I regard 'typo' as less formal. There are contexts in which I would only use 'typing error'. Perhaps that's the kind of difference your daughter has in mind.