Jamie I may not have crossed the 'River Mississipi' but I have crossed the 'Mississipi River' just seems you have the names backwards over there!!!!
Actualy the word River in front denotes the difference between the River Thames and the Thames Estuary which of course is at the end near the sea. Further upstream, where it is still a stream, it is just the Thames. So you have; a young river, a mature river, an old river and the inclusion of the word River tells you which bit your in/on/near.
A lot of our rivers have estuaries at the end and they tend to be tidal whereas the river part isn't so it is an important differentiation for sailors. If you were talking about the river's course you might talk about the Thames Valley which covers it from one end to the other.
Also some of our rivers share their names with places so you could talk about Hull, which is a City, or the River Hull which the City stands on both banks of and also bridges. It avoids confusion to be more specific - See its all in the evolution of history, which we have a lot of
