The main difference between using aesthetics and esthetics is where they will be used. When you say aesthetics, this can refer to a branch of philosophy that people are studying. This particular branch focuses on the appreciation of beauty. For example, you might say that you like the aesthetics of the sculpture. This means that you like the beauty of the sculpture especially if it is something that is completely unique.
When you say esthetics, this means that you are referring to someone who is a specialist in skincare. This is someone who would be able to provide a wide range of services that are meant to improve the skin. Take note of the different skincare services just to be sure.