expressions don't have an equal sign and equations do
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I don't really think there is a difference, but maybe the difference is that in pre-algebra you learn and practice the basic foudations, like the different steps to solve an equation, and what is a equation, and stuff like that. Begining algebra is where you practice actually solving the easier types of algebra equations.
A harder version of algebra 1
Adv. is accelerated most likely.
Algebra 1 is a class/course that is on a higher level than Algebra.
Pre-algebra is essentially the basics of algebra. Algebra can be tough for many, so as a way to break the ice, you (ussually) take a pre-algebra course before algebra itself. Algebra is a very large subject, spanning multiple years of courses in most (if not all) schools. After pre-algebra, you have to work with many algebraic equations in the later courses, which cannot be understood without learning the basics that are taught in pre-algebra.