Here are some examples:
Problems 18: Maximum path sum I and 67: Maximum path sum II are the same puzzle, just at different scales. It is possible (but needlessly slow) to brute-force problem 18 by checking all 16384 paths. However, problem 67 has 2^99 possible routes, making it completely infeasible to take the brute-force approach. Of course, both problems have very efficient solutions that should run within a fraction of a second on any basic PC.
In a similar vein, another nice set of problems are 81: Path sum: two ways, 82: Path sum: three ways, and 83: Path sum: four ways.
Solving these five efficiently involves applying concepts like this: