They are synonymous. But, I think the word gym is older and refers to more of a school-like environment in the days before we had real adult membership gyms. Don’t forget, in the States people rarely went to private gyms (except maybe the YMCA) in the 50s and 60s, or earlier. The fitness center trend for adults got really started in the 70s. I think the term “fitness center” just became a dressed up word for gym at that time.

They are similar. I think gym is more specific about hard workout and training. And fitness is more about reducing fat to get a lean body.

Depends on context. The area in a school where the kids go to play basketball and do other physical things is always a gym. While I feel like a gym denotes more weights and a fitness center has more cardio (treadmills, etc) and might cater more to women, anyone that says they are going to work out always says "I'm going to the gym." The words have become pretty interchangeable, except in a school context like I mentioned.

They are the same thing. You would hear "gym" spoken more commonly, but a gym might call itself a "fitness center." You would probably always say, "I'm going to the gym."