You are correct, and I don't think there is much more of a collaborative than a competitive spirit between the various organisations. However, it is not only the model that is important in weather forecasting, but also the data assimilation system. This is an active area of research with a lot of "art" involved, or, perhaps more realistically, very educated guesswork, about how best to set up and tune any given system. ECMWF have been developing their 4D-VAR system in the atmosphere for quite a long time now, and it is very successful. I think the Americans might be using an ensemble Kalman filter, which has a whole range of tuning issues, so that will probably take them a while to get right. There is an interesting paper that interprets the effects of these two systems that I still don't fully comprehend [0]. The Met Office also uses a 4D-VAR system, but I'm not really clear on the key differences between it an the ECMWF system.
Another important topic is coupling ocean and atmosphere models, and how you handle the data assimilation across your coupling - again, this is an active area of research, with lots of subtlety.
Of course, the model dynamics and physics are also very important, as is the resolution. This last issue is one of the places where bigger computers have really direct benefits, the other being increasing the size of an ensemble.
It's also important to realise that the various systems tend to be better suited to certain things, so whilst the ECMWF system is better in key global metrics, it doesn't (on average) provide better forecasts of all quantities in all situations.
[0] http://onlinelibrary.wiley.com/doi/10.1034/j.1600-0870.2001....