What is the best organization for a younger (GWOT) Veteran to join, American Legion, VFW, or something else?

I'm not a member of either, but I've been in a number of them. It's always the people that make the difference, not necessarily the organization. Some units are more political than others. Some units are very big into community service. Some units have a motorcycle group. Some have bars. Some are restricted and others are open to the public on a schedule. Bottom line, it will be whatever environment you feel comfortable in and that you get some return on your investment of time and effort. Up where I live VFW and AL are very small and both are run by a few who want to dictate how everything is done. They have no idea how to hang up the rank. So I'd recommend neither here. Results will vary elsewhere. Just start showing up at what's within reasonable driving and check them out. There's no rush in selecting.

It depends upon the post, not the organization. I joined both the VFW and AL, and attended meetings. Neither worked for me so I switched posts. On the second attempt I found what I was looking for (a group dedicated to helping/serving veterans). It happened to be the VFW. It could have been the AL. But I allowed my membership to lapse in the AL and devoted myself to the VFW post I discovered.

I think CPT Jack Durish has said it well. I can't say I felt as welcomed by the VFW early on and did better with the American Legion. I think the GWOT vets should be rubbing shoulders with the vets of previous conflicts. There lessons to be learned, stories to be heard, and is a great way to meet people from all walks. Service is rare and I think being around like minded warriors is important. The name on the sign out front is less important.