They're both Sinitic. They all have the area of meaning 氵short for 水 meaning (running) water. In short it means hot-water bathing.
(沐浴 (bathing) is made of 沐 meaning rinsing down the hair of the head -- 木 represents the sound, but it comes from the meaning of the look-alike 尤 dog element of 尨 meaning a shaggy dog and referring to its drooping fur/hair. The secondary part 浴 means pouring water on the body, 谷 coming from the shape of the container's resemblance in profile to a valley.
湯 has the element 昜 meaning a sun extending upwards aboveground. It has to do with the confusion of whether it means water stirring up when it boils in soup (as in Chinese, I hear), or hot water including the sort that springs forth from the ground (as in Japanese, though they often but not always write the latter sense in the native script).)