Contemplate conveys deep mental thought. It usually would imply that the item being contemplated has complexities and many potential outcomes. Ponder conveys thinking about something in a light-hearted and sometimes emotional way. It is more akin to dream than think. I am contemplating the advantages and disadvantages of selling my home. I am pondering about life in the city versus the country. While you could use either word in either sentence, the first conveys a very serious look at the financial and personal costs and benefits. The second conveys someone that is dreaming about what life would be like in either place, but they are not currently truly thinking about moving.

It's difficult to discriminate between these two words. It's possible that contemplate is a more serious word. "Ponder" sounds like a word for sissies.

I would say that ponder is more neutral minded. and when you contemplate then it is a problem that is a bit more serious.