I have to admit that I thought there was more of a difference between persuade and convince than there actually turns out to be, but there is some interesting background that we can look at too.
To convince someone of something means to cause them to believe firmly in the truth of something, or – aha – persuade them to do something.
And to persuade someone to do something means to cause them to do something through reasoning or argument; to cause someone to believe something; to provide a sound reason for someone to do something.
So, all very similar. But here’s the interesting bit, and it does come across a little bit in the above explanations … Traditionally, convince is used in situations in which someone’s belief is changed but no action is taken – “I convinced her to believe in the tooth fairy”, whereas persuade is traditionally used in situations in which someone’s belief is changed and action is taken – “I persuaded him to buy the red trousers, even though he usually wore black”. Feel free to keep that distinction going if you want to!