Okay, here's my shot:
Build:
Pros
1) can design own cabinet. You can make a slim cabinet for an LCD and have it take up less space, or do something batshit crazy creative
2) the satisfaction of having built the thing from scratch
3) a good excuse to buy a bunch of power tools if you don't already have them
cons
1) can get expensive, especially if you don't have the tools
2) takes up more space then re-purposing a cab
3) takes time to actually put together
Old Cab
Pros
1) Less wood working
2) you might get lucky and find one with a working monitor
3) possibly cheaper
4) less time - if you can find a cab in decent shape
Cons
1) can be frustrating to find one for a decent price in decent shape
2) you are stuck with the dimensions of a classic cab, would probably look retarded with another monitor in it
3) possibly stuck with some monitor work
Here is the deal. Most people will say "get an old cab, stick a tv in it and you're done!" But here is my 2 cents. Arcade cabs are 20+ years old, and most of them spent most of their lives getting the crap kicked out of them at the local 7-11. In other words, MOST cabs I have seen in my neck of the woods that are at all affordable are ugly and beat up. That being said, it takes a lot less tools to mend up an old cab, but you are probably looking at a lot of sanding, refinishing and stuff like that to get it to look good.
Then you have to deal with the monitor issue. Old cabs had 4:3 monitors in them. A widescreen monitor in an old cab looks like crap. Some are nicer than others, but they all look like crap, the widescreen throws all the proportions off. So you need to either get a arcade monitor, which is getting harder to find, or put an old tv in there, which brings up an entirely new headache of finding the RIGHT tv, (S-video, power return, input memory)
Here is my most important advice to you, no matter what anyone says, or whether or not you care about my other points, treat this one thing as a fact: A cabinet should be built around the monitor. Look through project threads and crap mame, and all the pictures of mame cabs. The ones that don't have the monitor match the cab all look like garbage. So if you get an old cab with a 25 inch monitor, then you should put a 25 inch tv or 25 inch arcade monitor in there or it won't look right, you will have huge/awkward bezels in there.
Really the true answer lies in what you want to do, what you want the cabinet to look like and what is available in your area. Some people on this forum swear by using old cabs, saying how they got their so and so cab in mint condition for 8 bucks and a box of little debbie brownies, but that just isn't the case in my market. Plus, in my situation, my cab is going in my home office, so I couldn't have some huge old beat up arcade cab in there, so for me it was a no brainer to build a slim LCD cab.