These are not two names for the same platter. The centre of gravity is different.
The word comes from a butcher's shop
Charcuterie is French. It joins Old French char (flesh, now chair) with cuit (cooked), plus the -erie ending that marks a shop. Literally it is cooked flesh, and it originally meant the shop that made and sold it, along with the trade itself — a charcutier — dealing above all in pork. In English it is usually said shar-koo-tuh-ree, with the weight in the middle; in French it is /ʃaʁ.ky.tʁi/. Which is to say the word never had a place in it for cheese at all.
The difference is what the board is built around
A charcuterie board is built around cured or smoked meat, with cheese in support. A cheese board is the reverse: cheese leads and everything else is there to hold it up. That changes the order you shop in. For one you settle the meat first and then look for cheese to go with it; for the other you settle the cheese. These days most boards carry both and the line has blurred, which is nothing to be pedantic about — but it is worth deciding which one is the centre before you start. When everything is the star the board simply becomes busy.
When meat leads, choose cheese like this
Cured meat is salty and rich. Put an equally salty cheese beside it and the salt doubles up, so people stop after a few bites. Give one place to something mild and soft instead: prosciutto with fresh mozzarella or burrata is exactly that principle, worked out long ago. Add one firm cheese and you get a change of texture as well. The Spanish version — jamón with manchego — is built the same way. Don't stack salt on salt, and you will not go far wrong.
The ones that hold their place when meat is the centre:
When cheese leads, one meat is enough
Going the other way, the trick is restraint with the meat. Two or three cured meats will blanket the board in salt and fat, and the differences between the cheeses stop registering. One kind, sliced thin, not much of it — then it does the useful job of resetting the palate as you move between cheeses. Keep it away from the most pungent cheese on the board, too, since those two will only cancel each other out. If the empty space bothers you, pickles and nuts will use it better than a second salami.
Can it be a charcuterie board with no meat?
Strictly, no — the word is the meat. Yet vegetarian charcuterie board is a phrase people genuinely use, and there is little point arguing about it. The real question is the practical one: take the meat away and something has to fill the gap. What the meat was providing was salt, richness and something to chew, so hand those three out separately. Olives or pickles take the salt; nuts take the richness and the chew. That works better than simply adding another cheese.
How much for how many — the sums don't care what you call it
Charcuterie board or cheese board, the arithmetic is the same: 30 to 50 grams (1 to 2 oz) of cheese per person as an accompaniment, 100 to 150 grams (3½ to 5 oz) if it is standing in for the meal. Aim at the lower end when meat is sharing the board, since it takes up part of the same appetite, and at the higher end when there is none. Either way three to five kinds is the range, and that does not grow just because the guest list does.