In general, these mean the same thing, but there important differences in usage in practice. For a general definition of tone in a photographic context, see What does "Tone" mean? — and just to save clicking, note that it has two common uses:

• the overall lightness or darkness of an area of an image
• the color of all or part of the image, usually in relation to its warmth or coolness

So, from one point of view, particularly when used in speaking about photographs, one simply might mean that. But skin color in humans has huge cultural and historical significance, and in particular the word "color" has implications of race (and often with color words used to describe ethnicity without actual consideration of literal color). It's therefore very common to use "skin tone" as a desired-to-be-less-loaded way to talk about literal skin color.

Particularly, "skin tone" is widely used in the cosmetics industry, which in general just wants to sell huge amounts of product to all people without getting into politics. In fact, a Google search for "skin color" returns results about genetics, race, and culture; while "skin tone" gives pages and pages and pages of results about makeup. And if you add "photography" to those searches, for "color" you predominantly get articles about the science and politics of race as related to photography (like this one about how metering is designed for caucasian skin), while "tone" is mostly stuff like using Lightroom's color curves.

So, while it might be tempting in the abstract to use "skin tone" to mean lightness or darkness and "skin color" to mean hue or chromaticity, in practical use that's unlikely to be clearly understood.