I can't speak of the parts interchangeability, but I made my purchase based on my experience with an 034. I learned on an 034. I was led to believe it died (which I now have it back, and it's going to cut again) so a few years later I bought a 361 thinking that it was the closest thing I could get to the 034. My personal opinion is that the 361 is a better saw. By a huge amount probably not, but still better overall. I really enjoy running either, and will probably just use the 034 as a backup when it's in fighting shape again. Around here you're limited to Stihl if you want a full service dealer and a professional saw so for general use/residential heating I like this size. Also I had a 18" bar on the 034, but the 361 is running a 20".
EDIT: But then again the 034 lasted through my grand-father, father, and me (the previous two owners didn't quite take care of it the way I did). The 361 hasn't really had a chance to prove itself yet.