I have had 3 Andersons (one classic, one Drop Top and a Cobra) over the years and also 3 Machetes with different setups. To me (!) they feel, look and sound very different. Both brands are outstanding and custom builders I'm sure they can build you anything you want. So all that I say is more related to the 6 guitars I had and possibly not to the brand.
Tom Andersons are perfect modern guitars. Optimized to the max. Build in perfection. Everything is very well balanced. Each tone, each string sound perfect, etc. The guitars feel like they are machine produced (I mean that in a good way) as they are perfect in every aspect.
The Luxxtones I have/had are all perfect rock machines. They sound and feel very organic. The pickups are probably the best I ever played. Really they are amazing! The guitars clean up perfectly. With my Machete and a good amp, I can do (with the volume pot and a HS setup) everything from Hendrix to Lynch/Jake E Lee/Van Halen in perfection. They feel like perfectly done hand made guitars. (keep in mind I only have experience with Machetes, can't comment on their other models at all; some of the things I like might also be a testament to the Machete model). The necks feel amazing (esp the oiled). All three I had were very different sizes, but everyone was outstanding. And of course, the looks are very special too!
Be warned, stupid car metaphor, but it's so hard to describe different feels: If you prefer something perfectly clean and optimized like an Audi, go for the Tom Anderson. If you love the rawness, aesthetics, feel and power of an old Mustang, go with the Luxxtone.
It all comes down to a matter of preferences in my opinion, neither is better than the other. Both are absolutely highest end guitars and you can't go wrong with either. Both builders are amazing to deal with too btw.
My personal #1 guitar is a Luxxtones and it let's all my other guitars collect dust or been sold. For me the perfect guitar! But again: that is a matter of preferences and taste.
