I want to go swimming and there is a private swimming pool close to where i live.
One is a membership and costs money. The other may or may not require membership but costs less, if at all, money.
Private swimming pools are a part of private enterprise, that is they are owned and run by people, private entrepreneurs, and usually have memberships, or at least entry fees. These pools are generally higher maintained and more modern than public swimming pools, as they can afford the upgrades and are secure in knowing that their patrons may treat the place and equipment with more respect, knowing they pay for it.
Public swimming pools are generally owned and run by the government or the local council. They may or may not have membership, although if they do it tends to be cheaper than that of private swimming pools. They also generally charge an entry fee, however this entry fee is once again cheaper than if you were to go to a private swimming pool. These pools are not as modern and not so often renovated and maintained, purely because patrons may not take as much care with the place and the equipment, as they may not have paid directly for these upgrades.
A private pool requires a membership fee and usually a yearly maintenance fee as well. It is owned and run by an individual or an organizational body like a pool association. People who are not members can not usually enter, although some private pools will allow guests.
A public pool is just that. It is open to the public by paying a daily admission fee at the gate. Season pool passes are usually available as well. It is most often owned by a municipality, run by an appointed board, and supported in part by taxes.
I disagree that public pools are by definition less well kept. If a community has the tax money available, the pool is often quite nice. In my local area, the public pools are much larger and better maintained than the private pools. Private pools struggle because the financial burden is solely on the shoulders of the membership, and many people can't afford it these days and are giving up their pool memberships.