Usually the odometer only starts recording when you have the timer on. It's the same with my Polar S710i. Well, all these bike computers with heart rate monitor do let you check in on your fitness. Assuming the rides are the same route, wind and environmental conditions are the same, a increase in speed and/or decrease in heartrate would mean increased fitness. The cheaper method would be just using a heart rate monitor and timing a loop that you usually run, a faster time and/or lower heart rate would mean a increase in fitness too.
You can check out the SRM, which I am pretty interested in right now. Otherwise the Vetta would be good, the one I'm using is Polar S710i which is pretty good for me, and the last one I know of would be ciclosport.