The compressor on my 9 year old rheem a/c unit burnt out. Will changing the compressor on my rheem ac unit harm the unit? Does it make more sense to change the whole unit?

If its under waranty yes. If not a lot of the manufacurers only offer a 1 year warranty on compressor change outs. This means this time next year you could be high and dry if it goes out again. If it's a r410 system it's a mater of just changing the out the out side system. If its R22 you will need to change out the outdoor unit and the metering device in the evaporator coil at least.