I have heard both, and to my ears the Rega is by far the better DAC. The Burson did not impress me at all with red book CDs, but it was much better with hi-rez files. I only heard the Rega with regular CDs, and I find it to be the best I have heard so far (but have not heard that many).
On another comparison with the Bryston DAC, I also preferred the Rega by a significant margin. That audition was on a very revealing system that included Vandersteen Model 7, an expensive AR preamp and Bryston amps. The Rega was very close to good analog sound and was free of most digital artifacts. I borrowed the Rega for a couple of days, and my opinion has not changed. As with all digital electronics, make sure the unit is broken in and has been on for about a day before a serious audition. The Rega sounded a bit harsh when cold.
On another comparison with the Bryston DAC, I also preferred the Rega by a significant margin. That audition was on a very revealing system that included Vandersteen Model 7, an expensive AR preamp and Bryston amps. The Rega was very close to good analog sound and was free of most digital artifacts. I borrowed the Rega for a couple of days, and my opinion has not changed. As with all digital electronics, make sure the unit is broken in and has been on for about a day before a serious audition. The Rega sounded a bit harsh when cold.