Both Weeks and Mapp have to do with the "exclusionary rule." That is the legal principle that says that evidence can not be used in court against you if it has been gathered illegally.
In both Weeks and Mapp, the Supreme Court declared that evidence used against the defendants had been gathered illegally. Therefore, the evidence could not be used against them.
The difference between the two has to do with who the Supreme Court rulings applied to. In Weeks, which was decided in 1914, the trial was being held in a federal court. So that decision said that the FEDERAL government couldn't use illegally gathered evidence. However, it was still okay for STATE governments to use that kind of evidence.
In Mapp (decided in 1961) the trial was being held in STATE court so the exclusionary rule didn't apply. However, the Supreme Court decided in that case that, from then on, the exclusionary rule WOULD apply to states as well.
So -- the similarity is that they both upheld the idea of the exclusionary rule. The difference is that Weeks said it applied to the FEDERAL government while Mapp said it applied to the STATE governments too. So now it applies to both.