What are differences between the Euclidean and Hilbert planes?

You mean the parallel postulate. Yes, the set of axioms which a Hilbert plane satisfies does not include the parallel postulate, nor the negation of the parallel postulate, as well as omitting axioms concerning intersections of intersection of lines and circles that are used to axiomatize Euclidean geomety. For more details, check out page 97 and the pages before that in the first reference below, or, a bit more accessible, the second reference.