How do I extract Month and Year in a MySQL date and compare them?

There should also be a YEAR().

As for comparing, you could compare dates that are the first days of those years and months, or you could convert the year/month pair into a number suitable for comparison (i.e. bigger = later). (Exercise left to the reader. For hints, read about the ISO date format.)

Or you could use multiple comparisons (i.e. years first, then months).