Is the DA-20 still called the Katana? Thats what it was called when I did my PPL in it! Anyway Eclipse/Katana they are both DA-20. I did some limited time in cessna 152/150s and a couple of hundred in the DA-20.
In my opinion the Katana is slightly more demanding to fly. I found the cessna a more stable a/c for practising basic handling - the Katana is basically a glider with an engine at the front but the resulting L/D ratio means if you trim it properly when practising your PFLs/Glide Approaches you get plenty of time to pick a field and judge your distance etc. I also remember the Katana being very slippery when practising flapless landings - you really have to keep an eye on your IAS as it doesn't want to slow down! One of the big plusses of the DA-20s I trained in was the electric trim on the stick - it made life easier and it was very instinctive once you got used to it. I havent seen that sort of trim system in a cessna.
Regarding the stick/yoke it doesn't matter a bit, it might take an hour or so to adjust but I found transitioning between the two straightforward. You will find it's the same when you switch from Boeing to Airbus later on!