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.

No experience with the AL but I'm a life member of the VFW. Can't speak on the VFW nationwide as I've only been at one post, but as for my local post I have a lot of respect for them. They're very active in the community. They do a great job at helping veterans and the community in general. I've had the opportunity to network with a few of them, particularly one key member of my post is my county commissioner.

Honestly, it depends on the post. I have belonged to both but am now a lapsed member of the American Legion, and active in my second VFW post. The first VFW post didn't fulfill my need. It didn't provide the opportunities I was seeking to serve other veterans. None of the AL posts did either. I'm sure my experience would be different if I lived in a different area of the world

Depends on the post. However, the DAV is best with helping with divisibility claims.

This is just my opinion, and I know it's an unpopular one but it's how I truly feel. I don't support either organization. When the government was in the process of repealing DADT, both the Legion and the VFW took the stand at capital hill and fervently fought against it. They threw every single gay, bisexual, lesbian, and transgender soldier under the bus and never even said they were sorry for doing so.

Then they did it again during the latest controversy with transgender service members under serving under President Trump. They don't have my back, they don't want anything to do with service members like me. If I walked into a AL or VFW lodge presenting as a female then I'd be lucky if they even let me sit down at the bar and order a drink. My service didn't matter one iota to them, so I refuse to fund them in any way.

""We feel strongly that the current policy has served the U.S. military well for 17 years and it would not be wise to make a major cultural change. Allowing homosexual members to announce their preference within the ranks could jeopardize the unit cohesion which is so essential to military success." -American Legion National Commander Clarence E. Hill

That's still their official policy too, according to this statement on their website:
https://www.legion.org/legislative/point-papers/90629/dont-ask-dont-tell-policy

As for the VFW:
"Repeal advocates are focused primarily on pushing a social agenda about individual rights. National security, unit cohesion and morale are the furthest things from their minds. This social experiment could spell the end of America's all volunteer military, which is not a price this nation appears to be willing to pay." -National Commander Richard L. Eubank

Depends if you like to drink... my friend liked to go to the AL post to pick up women.. if your combat deployed ... join both.

I am a life member of the VFW but have never been to a post. I recommend finding out more about what the local posts are involved with.