有病 sounds a little bit more serious to me than 生病, but that might be completely wrong (I'm not a native speaker). However "有病" can also mean "to have a mental problem" or just note somebody's crazy behaviour.
我生病了 - I got sick - emphasizes that you are sure you got sick. 我觉得不太舒服 - I think I am not well - is still suspecting, not entirely sure if you got sick, but more like a feeling.
了 = change in situation