Improve - Something is not really acceptable or at the level we want. We can improve it to meet our standard. enhance - something is "ok", but we can make it better, nicer, more attracctive, etc. If I improve health care, people can now go to the doctor and get medicine and health care (before they could not do that or it was difficult.) If I enhance health care, people already have good health care, but now they can go and learn about healthy lifestyles, take free or low cost exercise classes, have a nurse come to their house to talk about prenatal care with pregnant women, etc.)

Good question and it depends on the subject and context. They basically mean the same thing but usually used in different scenarios. For example you can improve in a sport by practicing. And also you can take performance enhancing drugs. Its not normally said that you enhance your skills in a sport by practicing, but that would also be correct. Enhance has somewhat of a connotation of "artificially". you improve your appearance by exercise and diet. You enhance your appearance by plastic surgery(cosmetic surgery). I hope that makes sense.

I'll make an example using food. In the kitchen, if you somewhat fail or burn an omelette,you can try again and improve. Second, you made an omelette and its okay, but you can enhance it by adding flavor, to make it more delicious.