[susan53]

I've looked through about 200 sample sentences and can't find any difference- as long as you're talking just about the forms convince/persuade as active verbs and not related forms like convinced (adj) and so on.

You can both convince/persuade someone to do something and convince/persuade someone that something is true. Maybe there's sometimes a slight difference in meaning : in some (not all) of the examples convince seems to be preferred when the writer is talking about using a logical argument, while in some (not all) others persuade seems to have been chosen to express an emotional argument.

I also considered that maybe if you were convinced to do something you always believed in it, while you might be persuaded without really believing it was a good idea. But Google
I/He/She (etc) persuaded /convinced me/her/him to do it against my (etc) better judgement
and both verbs pop up. So that's out too.

I reckon they're close to synonymous.