Although they are similar, they are really quite different. A health club would normally be more luxurious than a gym. It would have more facilities. A gym can just be a room with some gym equipment in it. A laundromat is a place that has self-service washing machines. It may sometimes have an attendant, but you would normally expect to be able to do your own washing. A laundry is much more general. It can be used to refer to the room in your house that has the washing machine and tub in it. It can also be a commercial laundry that you take your washing to, the people ther wash it, and then you pick it up later. Some places can be both a laundromat and a laundry.

A health club will probably have a gym but also other things like a sauna, maybe a pool. A laundromat is a place where you pay to wash your clothes and is an American word. It is the same a launderette which is the British word although British people might also say laundromat. Laundry is the word for the clothes you want to wash.