For our main summer vacation last year M. and I chose Sardinia. Since we have already been to its closest island neighbour Corsica, we expected to find similar things there. Well... things were quite a bit different but definitely not disappointing.
Since these two islands are literally right next to each other, one would naturally expect similar landscapes. With the exception of its northern part, Sardinia is not very much like its northern neighbour. Sardinia (with the highest peak of 1834 meters) is hardly a match for over 20 mountains higher then 2000 meters on Corsica. If climbing is your thing, I suggest you choose Corsica. For hikers both islands offer more then enough options.
For doing a round trip on either island, one should consider island size. Corsica is 2.5 times smaller then Sardinia. I guess three weeks are just enough for easily making it around Corsica. We tried to circle Sardinia in such a period of time and it was not such a great idea. We ran out of time and we had to quickly drive through many beautiful areas, completely skipping some of them.
Both islands have many stunning beaches. Sardinia is larger and has more of them. If sandy beaches are the kind you are looking for, Sardinia has more of those to offer. This doesn't mean there aren't any on Corsica. Sandy beaches might not be all that frequent on Corsica, but if you hate getting that fine sand everywhere, you might actually like this fact.
I also got an impression there are more historical sights and museums worth checking out on Sardinia. Corsica has a few museums, lots of menhirs and Genoese towers (mostly closed for public), but all of that can hardly be a match for countless displays of history on Sardinia.
Decisions, decisions, decisions... Corsica vs. Sardinia, nature vs. history, adrenaline vs. leisure, French vs. Italian and so on and on... The choice is yours.
Which ever you choose, you will not be sorry. If you can afford it, visit both. Just remember to take enough time to enjoy the trip and try not to rush from point to point. Keep in mind that things between planned destinations can be well worth your time as well. Actually just those unexpected jewels can be most memorable points of such a trip.