Examples taken from merrian-webster defition of convince include both the use of of and to:
He convinced me that the story was true.
They convinced us of their innocence.
I managed to convince myself that I was doing the right thing.
We convinced them to go along with our scheme.
I was unable to convince her to stay.
However, according to thesaurus:
The use of convince to talk about persuading someone to do something
is considered by many British speakers to be wrong or unacceptable. It
would be preferable to use an alternative such as persuade or talk
So it would seem to be just another BrE/AmE usage issue.
The definition in the Collins English Dictionary also points in
- (may take a clause as object) to make (someone) agree, understand, or realize the truth or validity of something; persuade
- Chiefly US to persuade (someone) to do something
Besides this usage issue, there is also, as JohnJamesSmith said, a difference in meaning.
Here is an interesting discussion at WordReference, which gives a very good example, concluding that there are cases where the replacement of convince for persuade just for the sake of correctness, can eventually have an actual degradation in meaning. Let me quote the post:
I actually feel like there's a meaningful difference between convince
someone to do something and persuade someone to do something, at least
in certain contexts.
Since convince, as you say, geostan, involves changing someone's
beliefs or opinions, I feel like that semantic property holds even for
the convince someone to construction. Let me give you an example:
Two friends, Bob and George, get caught stealing from a store. Bob is
always getting into trouble, whereas George is generally a nice boy.
When questioned by his parents, George says, "I know it was wrong to
steal, but Bob convinced me to!"
In this context, convinced implies, at least to me, that George
changed his beliefs, at least temporarily, in order to steal. That is,
for that brief moment, Bob managed to make George believe that it was
okay to steal, or that he could ignore his conscience/beliefs.
Persuaded, on the other hand, would imply that George stole knowing
(and still believing) full well that it was wrong.
It may not be a great example, but if you consider the adjectives
convincing and persuasive, I think the difference is there. We could
say that "Bob was convincing" means that Bob was able to convince
George that stealing was okay, whereas "Bob was persuasive" simply
means Bob was able to get George to steal, without having him change
So, I would conclude that in most cases it is a matter of BrE or AmE use, but you should always bear in mind that you can not always replace convince with persuade, because both verbs have slightly different meanings.