Congratulations on your great offers! I agree with everything that has been already said by "cheetah", but want to add a few thoughts and insights that I received from several alumni that worked at both Fletcher and SAIS before (students and non-tenured faculty/postdocs)
- Fletcher is generally known for having more "research-focused" professors. Their close connection to Tufts allows Fletcher to have quite a few research projects that are also being recognized outside the policy world. The Fletcher-PhD also seems to be relatively academically renowned and the connection to some Harvard grad schools is still close. I know a couple of people that focused on more of a research path at Fletcher and wrote books together with faculty to then pursue a PhD in a top IR-Grad-Program. However, on the student level, the same applies to SAIS - you will find some alumni that got into top-20 PhD programs after a master at SAIS and some work experience. Just the faculty at SAIS does not seem "as" academic.
- Fletcher is in policy circles more known for their close ties to the State Department. SAIS however is more known within the DC establishment, especially in think tanks. You have to know what you prefer. I also feel like that Fletcher attracts just slightly more international students with the goal of getting into international organizations, like the UN etc. If you want to establish yourself in DC circles, however, SAIS is definitely the better choice by far! For everything "more international", I would pick Fletcher.
- For some reasons, some Fletcher-insiders I know are reporting that the department would be on a slight downfall at the moment. I don't know any specifics, but received that information from several alumni that claimed that the change of administration within Fletcher in the last years did not do the school well. SAIS is also changing quite a lot currently - I just cannot see a clear direction there at the moment. If and when SAIS moves to the old Newseum in DC, this might be a big change to the general environment of the school.
--> I believe in the end it should come down to the location (and finances). SAIS gives off very professional vibes, so you have to be "ready" for it. Fletcher still has some college campus feeling to it and is situated within a quieter environment, which I would honestly prefer. At the same time, SAIS is in DC, which is a big plus, but only if you aim for DC only jobs and the respective networking in the future. Fletcher won't close any doors related to DC, SAIS just might open a few more. The community at Fletcher must be incredibly nice though - I hear that quite often! and honestly more than in relation to SAIS.
(I would not go to GW to be honest. Fletcher and SAIS are just in a different league)
Most success to you and good luck for your decision!