Well, an amusement park is just that. A series of exhibits strung together to keep people entertained. There is usually very little order to where things are placed (other then where there is room) and all theming if at all is pretty much minimal. A theme park has some kind of order. There is usually a theme of some sort that either carries the entire park (Like, Epcot, MGM, AK, BG, SW or US) or carries separate areas (such as MK or [shutter] IOA).
Theme parks almost always have only one central entrance and are designed with attractions that compliment each other being placed together. There is also usually a much higher attention to detail placed in both the attractions and customer areas. Amusement which were often started on a very small scale as a way to attract people to a part of town or to promote the use of public transportation around the turn of the century have been around a lot longer but most suffer from the lack of a master plan in their design. They started out small and expanded over the years/decades wherever they were able to find room regardless of whether two attractions would fit well together or not.
Today, because theme parks are regarded as higher quality, many amusement parks have begun to add theming to their attractions and to pay more attention to landscaping and other environmental conditions such as music and unique architecture in an attempt to blur the like between theme park and amusement park. IMHO, for instance, IOA is a highly themed modern amusement park which while top of the like for an amusement park, falls short in many of it's areas from being a good theme park. On the other hand, US, run by the same people is a full theme park that carries it's theme throughout. (not that I'm saying it's as good as Disney, mind you
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I'm sure I've missed a few things in this post that others on the board will probably be able to add but this should provide you with the basic concept...