when you are trying to convince someone , you try to make the person share and agree with you said by laying out facts,arguements or evidence
when you are trying to persuade someone , you are doing the same thing except that you are more likely to be appealing , begging , playing with feelings to bring him over.
From the Net :
To urge is to "To encourage forcefully", "to recommend earnestly" eg I urge you to stay, please