To many, these two are synonyms, but there are shades of difference. Someone might be persuaded, while at the same time, not convinced: She persuaded me to do it, but I'm still not convinced it was right. When something or someone persuades us to act, it might be by using reason or logic, but it could also be by using force, lies, or guilt.
Convince refers to an unforced change of mind and heart that precedes action. We consider the evidence, and if it is strong enough, it convinces us and changes our perspective.
In formal writing, convince never takes an infinitive, but persuade almost always does. You cannot be convinced to do something; you can only be convinced that something, or be convinced of something.