Endowed Assistant Professor vs Assistant Professor
There really is very little difference between the answer Chris Gregg posted and the answer for this question. An "endowed" assistant professorship is a position which provides funds for the group of the faculty member appointed to the professorship. It can be either salary, or perhaps additional funds that can be used to support members of the group. 

Usually, applicants are not hired directly into such positions; instead, they are hired to the department first, and then appointed into such positions after a few years. They often have names like "career development assistant professorship" which indicate that this is a transient award, not a permanent one. If the hiring advertisement actually specifies the position as an "endowed" assistant professorship, I don't think it really changes the hiring process, except that they're looking for the best candidate who satisfies the conditions of the endowment.