Between the American Legion and the VFW, which is the better organization to be a part of, and why?

Both are good, the American Legion has a larger recuiting pool since the VFW requires having served in a combat zone.

I've been a member of the Legion for about four years now. Currently, I serve as our Post's adjutant, and work regularly with the VFW. Every Post is different, as is every Department (state level); depending on location, number of members, activity level, and community involvement... experiences can vary widely. Those of us who've served in the last two decades have fewer "complications" based on the entry requirements for either (and with new regs passed this year, there are fewer for other veterans too). The old jibes about it being a place for card playing drunks are (in my opinion) a bit dated... and certainly not the "norm". I'd suggest you attend a couple meetings of both and get the "feel" for what's going on there, then decide if it's for you. I will say that younger members tend to find a warm welcome (and lots and lots to do)... as the Legion is trying hard to recruit new blood.

I would say that neither is better. Its more a question of which one is better for you. If you are looking for differences the major ones are 1. the American Legion has broader eligibility requirements (ie. any honorably discharged Veteran can join) whereas the VFW requires that you received a badge or medal for overseas service (they like to say combat service, but, strictly speaking, it isn't), 2. The VFW is more bureaucratic than the Legion. If you are a leader in a VFW Post you have to make sure you do the quarterly financial report, have an annual inspection with about 30 items on the checklist and adhere to a number of little rules as laid out in the VFW Manual. The Legion lets each post manage its own affairs and doesn't bother them too much. 3. Membership renewals and life members. In the VFW annual membership renewals are centralized so the member pays his due to National. In the Legion dues are paid to the Post and the Post Adjutant sends the renewals up to the Department (State) HQ for processing. The VFW has reasonable rate for life membership ($410 at age 31) whereas the Legion charges your current dues rate and assumes you will live to be 90. (i.e. If you are 30 and your Post's annual dues are $30 you will pay $1,800 for a life membership. No joke.)
Like I said at the beginning. The question isn't which one is better - the question is which one is better for you. Join both and decide for yourself.