The 1125 Series of saws which includes the 034, 036, and MS360, are great saws. They only made the MS360 for a couple of years from maybe 2000-2003, so not a ton of them are out there compared to the 034, and 036 which was made for more years. The 034 was made around from around 1985-1995, and the 036 was made from 1996-2003, I probably off on years, but just an estimate.
The 361 is has better anti vibration, than the 360, springs in the 361, versus rubber buffers in the 360.
Parts aren't interchangeable between the 360 and 361, but most parts from 034's, 036's will fit MS360's
Both have about the same power, the 361 feels a little heavier and I think has a little more power.