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")
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