What’s the difference between a board member and an officer?

The board of directors is comprised of the largest shareholders and/or of shareholders who have been elected to the board by other voting shareholders. Officers (CEO, CFO, CMO, COO, etc.) don't even necessarily need to be shareholders, though they usually are because shares of stock are generally part of their compensation package even if not already shareholders, but are employees the board of directors hires to run the corporation, people with the skillset needed to run the corporation. 

Just because someone is an owner of a business doesn't mean that someone is skilled at running that business or is the best person to run that business. Very often, that is very much NOT the case. That's why boards and officers. The board is the business owners or collectively represent the owners. The officers are employees hired by the board to run the business on the owners' (i.e., shareholders') behalf.

Say that I buy a coffee house because I think that coffee house can make me more money than anything else I might otherwise choose to invest in. The thing is, I don't know the first thing about coffee or running a coffee house and, frankly, have no interest in running a coffee house. So, I hire a manager with 10 years of prior experience managing highly profitable, independently owned, coffee houses like mine to run my coffee house, and to further incentivize that manager, I offer that manager some minority partnership in the business because then the manager will work harder to increase the value of the business and increase profits because doing so further enriches her, directly linking performance to compensation. In this scenario, I am like "a board member" (in this case, the only board member) and the manager I've hired is like "an officer."

Board member, as the name implies, simply means you sit on the board of directors.  Board memberships can be on a rotating basis, they can be appointed or elected by shareholders, or they can be earned with an ownership stake in a company.

An "officer" is an executive hired by the board to perform a specific function, the two most-commonly known are Chief Executive Officer (CEO) and Chief Financial Officer (CFO).