You don't need to compare your results to theirs to get additional information, as you can't have any DNA they don't have (unless, of course, there was a NPE [non paternal event]). Do you share the expected amount of DNA with each of your grandparents? If so, then you will want to work off of each of their tests to continue your research.
As to what to do next, you can look at the shared matches they each have on ancestryDNA and start seeing if you can further your tree by locating the common ancestors.
On Gedmatch, you can start looking at the matches that share the largest amount of DNA with each grandparent. It is easier to find the common ancestor with those, as they will be closer in time. I would start with those that share 20+ cM. If you don't have those, drop it down to 15 cM. You will want to write a standard email that you can send to matches to try and locate common ancestors.
How extensive is your family tree? You will want to get it back several generations in order to have a good chance at locating common ancestors. You need not only the names of parents of everyone, but also the names of all of the children and their spouses within each generation. It takes a long time to develop a good tree, but this is essential for success in using DNA to locate and/or prove ancestry.
You may want to use GenomeMatePro (free software) to help track your research: https://www.getgmp.com/