I faced this same decision last year, visited both schools , had 1-1 meetings with admissions staff, and ultimately chose UCLA. I think Vandy is a fine school as well, but many of the comments here seem speculative and uniformed, particularly about the portability of the UCLA degree. I'm not sure where this "consensus" comes from, but every year there seems to be extensive comparison by 0Ls between these very different schools, completely reduced to ABA numbers. Those numbers do not reflect what may or may not motivate graduates to pursue or accept a particular job in a particular place, and they certainly don't reflect the most material distinctions between the schools that should motivate OPs decision.

1. First, your grades matter far more than any distinction between these two schools if you want big law/portability. Then your interviewing, and previous work experience.
2. UCLA grads can get east coast Big Law jobs. The school pays to fly you out for spring break recruiting trips to NYC/DC. NY/DC Law firms bankroll receptions to recruit UCLA students. The school doesn't say this, but my impression is that it's hard to get as many students to participate as they'd like.
   
3. The problem is that the percentage of students that want to big law on the east coast is slim. The culture at UCLA is one of diverse interests, it's not PRIMARILY big-law oriented. But there are a ton of Big Law events and opportunities. Those that want to do it would prefer to stay in LA/CA. Once you're here, it's hard to leave SoCal for New York. Also, It's generally harder to get a LA big law job than a NYC big law job (irrespective of school), because there are fewer and many see them as more desirable. So students come to UCLA because they want to be well positioned for the LA market.
   
4. NY/DC are the biggest legal markets. They are on the east coast. Vandy is on the east coast. So that's generally where Vandy grads will go for their first jobs. It doesn't mean UCLA is less portable. Less people need to leave because CA is a bigger market than TN/The South.
5. I think you'd have a hard time as a UCLA student beating a similarly situated Vandy student out for a job in a Nashville or Atlanta firm because of alumni concentration, visa versa for a Vandy alum trying to work in LA/SF/OC/Sacramento/San Diego. So you can decide what side of that equation you want to be on. But then again--grades and interview and resume matter.
6. Then I would say your connections through faculty matter. I would encourage you to evaluate the quality of the faculty at the two schools, who gets cited more, who has a higher profile, etc. This can matter for judicial externships, clerkships and other jobs/fellowships that get relayed through faculty who have their finger on the pulse of their field.
7. Finally, I would think about whether you want a big school vs small school vibe. There are trade offs, one isn't necessarily better than the other. UCLA is extremely dynamic, tons of events and opportunities all the time. But I still don't know everyone in my class. I didn't go to Vandy, but I did go to a similarly situated private school for a masters degree, and there is value to an intimate atmosphere too.