In C1 and then copied down you could put:

=IF(VLOOKUP(A1, Sheet2!$A:$B, 2, 0)=B1, "Same", "Different")

If you'd like to note unique entries, too, then perhaps:

=IF(ISNUMBER(MATCH(A1, Sheet2!$A:$A, 0)), IF(VLOOKUP(A1, Sheet2!$A:$B, 2, 0)=B1, "Same", "Different"), "Unique")