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It would be better to try with simplification_routines(expression1-expression2)==0 instead of expression1==expression2 or simplification_routines(expression1)=simplification_routines(expression2)

The following seems alright:

var('x, y, n')
assume(n,'integer')
assume(n >= 0)
bool((((x-y)^n)-(-1)^n*(y-x)^n).expand().canonicalize_radical()==0)

It would be better to try with

simplification_routines(expression1-expression2)==0

instead of

expression1==expression2

or

simplification_routines(expression1)=simplification_routines(expression2)

The following code seems alright:to yield true:

var('x, y, n')
assume(n,'integer')
assume(n >= 0)
bool((((x-y)^n)-(-1)^n*(y-x)^n).expand().canonicalize_radical()==0)