Do I need a home inspector when my bank is having the house appraised?

Yes

A house appraisal is an independent evaluation of the current market value of a house or property. In general, the purpose of an appraisal is to set the current value so that a lender may determine how much it can loan the buyer.
A home inspector conducts a thorough evaluation of the houses major systems and structural integrity. Whereas the appraiser is typically working for a bank, the home inspector is working for you.