In both the approaches you divide a problem into smaller subproblems, solve them and combine their results to solve the bigger problem. This breaking up of the problem into subproblems can be several levels deep.

The difference between the two lies in the nature of subproblems. In divide and conquer paradigm, all subproblems are independent in nature and they do not overlap when you keep dividing problems into subproblems, each subproblem is required to be solved only once.

While in dynamic programming a subproblem solved as part of one bigger subproblem may be required to be solved again as part of another subproblem, so the very first time a subproblem is solved it's results are tabulated(stored in some efficient datastructure) and the next time it is encountered instead of solving it again, the result is simply looked up in the table and returned.