OK, socialized medicine is like this: there are a bunch of "government approved/contracted" doctors who perform services for a quoted price from the government. Usually, they are overloaded with work (cause the .gov is paying for it-->no incentive for people not to see the doctor when their pinky toe hurts) and usually can't take on additional patients. Doctors do have a finite amount of time to work. They cannot see patients 24/7. A private doctor, one who is not relying on the government to pay their salary, can see whoever they like. Their patient loads are usually less (since most people don't want to actually pay for services) and thus can see additional patients if they have the money to pay.
The thing that you said about having the money and still can't see a doctor is one of the "after effects" of long term socialized medicine. Akin to the system that is currently implemented in England (and has for some time now) medicine as a career field is just not as attractive to potential applicants, and thus the total number of doctors drops in time. When they get down to a point where the total "need" of the patients exceeds the total "working time" of the doctors, and the doctors are contractually obligated to provide services only to those being payed for by the government (government has to maintain market share) those doctors do not have additional time to see other patients, thus the scenerio you spoke of.