You're probably pulling back on the stick too much. The seaplane won't just rocket straight up into the sky, you need to bring it up into the air slowly. Pull back too hard and it just sort of stalls. Give yourself time to gain altitude slowly, you don't have to rush through this mission, there's plenty of time to do it. Hope that helps.
One of the great things about Vicy City (and what is making this mission frustrating for you) is that the vehicles handle realistically. You've got to go easy on that little prop engine.