Both are "accredited" in that both will meet the educational requirements for Texas MT licensing.
I went to The Lauterstein-Conway, it's a great school. It's the school itself that inspired me to become a massage therapist. People from outside of Texas who are involved in massage education know about TLC, it's "nationally acclaimed." The two founders, who still teach there, have a holistic approach to both massage and massage education. I highly recommend the school. Go to a visitor's night open house with John Conway, see what you think for yourself. (http://www.tlcschool.com/career_night.asp)
I've met MT's who are graduates of New Beginnings and they're as capable as any. Don't know much about the school itself.
The only one besides TLC I visited was Austin School of Massage and I didn't get a great feeling from it. Not to say the students don't go on to be good MT's, but the atmosphere just wasn't so welcoming to me. I heard it described once as the "Wal-Mart" of massage schools, a little bit cookie-cutter. They've changed things up since the increased hour requirement in Texas, adding on several "specialties" after the basics are covered. To me, that just means there's several options OF cookie-cutters
There's one more massage therapy school that I hear about in Austin, The Healing Arts Institute. I like the acronym, THAI. They seem decent, and turn out good therapists, but they were in the process of moving when I tried to visit it felt chaotic.
So, that's Austin! Definitely check out TLC. http://www.tlcschool.com/