As a current, outgoing chapter officer at my medical school, I am probably a bit biased but I think that AMSA is one of the organizations that is worth joining, especially if you are interested in getting involved in health care policy issues. I recently attended the National Conference which was held this year in Houston, and I was amazed at the number of medical students who take enough interest in policy issues (both at the national level, like universal health care issues and limiting resident work hours, and at the community level, like sponsoring health outreaches in individual cities) who were willing to take time out of their schedule to gather with other students, hear what was going on in other states, and rally for improved immigrant health care in Texas (this year's convention topic).
In some ways I think AMSA may fall trap to the bureaucracy that plagues large organizations, but maybe this is the only way to go to be taken seriously... I do know that AMSA also sponsors interesting summer electives for first year students as well as connects people to international health issues and rotations abroad. At our school, AMSA, one of the 'wealthier' organizations, tends to be very active in supporting grassroots efforts and community awareness activities.
This is probably a lot more information than you wanted to know about AMSA
and not much about the other 'big' organization, the AMA (which gets you JAMA free, a nice perk if you ever have time to read it). My recommendation would be to join both if you can... they both offer a lot of opportunities and connections if you take advantage of the organization, and both should give some nice freebies for joining (my year, we got a free Netter for joining AMSA and a free Stedman's for joining AMA).