I'm not sure there is an exact answer for this, much like if you asked someone if "indigo" and "purple" were different colors. From what I can see, 桃色 and ピンク are practically interchangeable.
This color chart shows both colors as different. ピンク is standard pink and 桃色 is closer to peach color. However, a 桃 isn't like a Western peach, it has many shades, many of which are like the Western pink color.
This chart doesn't have a 桃色 entry at all, only ピンク. You'll notice it has a ピーチ entry as well.
Color naming is not scientific, except for RGB, CMYK, etc systems. It's subjective, so much like "scarlet" and "red", 桃色 and ピンク appear to be interchangeable.