As has already been said, theres no reason not to use both, but I suppose its dependent on what it is you want.
If you want instant feedback on speed, avg speed, distance or whatever, then a cycle computer on the handlebars is the best - and safest way - of doing it I suppose. Easy mountable, and its just there at a glance whenever you may want that information.
If you want to 'track' more information, and log it for future back-reference, use an App especially if the strong likelihood is that you will carry your phone anyway. You wont be using it for much else whilst you are peddle beating, so may as well give it some job to do. What appeals to me about apps is that its logged data, so very easy to track your progression in quite a few ways - how fast did I take that hill/complete the circuit this time, how has my maintained avg speed improved this month, whatever.
Never used CycleWatch so can't comment; I've always used Mapmyride as it was the first one I came across and find it does exactly what I want. I do love - in absolutely the geekiest of ways - getting home, stopping the recording, by the time I've sorted myself out, jumping on the PC and going to the website and looking at all the splits and graphs on time/speed/climbs etc. Never had a failed recording either, touch wood.
In fact, at the moment my cycle computer is actually bust, and I aint too bothered about replacing it. Up-to-the-second current speed I think is rather valueless information (beyond the Weeeeeeee factor or on the odd occassions you may be looking at sprint speeds), and watching the miles creep up slowly in 0.1 increments doesnt personally do much for my motivation. I've just set MMR to tell me every 5 miles what my total distance, pace, avg speed etc is. Then I can adjust/push for the next 5 miles accordingly. Also a good method of pacing; if I hit mile 5/10 out of 60-70, and I hear that voice telling me that Im doing 3-4 minute miles, then I know that I need to back off a little or I wont last the distance.
