Most American local governments have both bodies, made up of appointed citizens, and they do different jobs. The planning commission looks forward: it prepares or reviews the comprehensive plan, holds hearings on rezonings and changes to the zoning text, and recommends to the elected governing body, which makes the final decision.
Many planning commissions also decide some things themselves, most often subdivision plats and, in some places, site plans or certain permits. Which ones depends on state law and the local code, so the same project can stop at the commission in one city and go on to the council in another.
The board of zoning appeals, called a board of adjustment or zoning hearing board in some states, applies the law to individual properties. It typically hears requests for variances, special exceptions and appeals of a zoning official's decision. Its hearings are quasi-judicial: decisions rest on evidence and written standards, and in many states board members should not discuss a pending case privately with either side.
The practical difference for an applicant: a rezoning is a policy argument made to the commission and then to elected officials; a variance or special exception is a case proved to a board against written criteria. In Charleston, South Carolina, for example, both bodies exist, with powers set by the state's planning enabling act.