The correct term is "Democratic Party". Some Republicans have begun trying to call it Democrat party, as they fear the name Democratic Party may indicate that it is somehow more "democratic" than the Republican Party, so they are trying to dispel this possibility by unilaterally choosing to call it by an incorrect name.
The party's official name is: The Democratic Party. Some argue that Gingrich omitted the "ic" for political reasons -- to more clearly separate it from the meaning of democratic (little d). The controversy is silly -- all that matters is that you know what party a person is referring to.
There is no difference. People refer to the Democratic Party as the "Democrat Party" in order to make insulting a statement. They are trying to say that the Democratic Party is not about "democracy" and so by saying "democrat" and not "democratic," they purposely make a distiction between the people and what they should be doing.
There is no difference. They are the same thing, it is just different terminology. Techinically and grammatically it should be callled the Democratic Party and the people who make up that party are called Democrats. This refers to the Democratic party which is the contrast to the Republican party whose members are called Republicans.