@dcsccccc - I misremembered the scope of the atavistic legs; they are only present in whales, not dolphins.
First, let’s take a look at the evidence for atavistic legs (not fins) in whales. I will be quoting several reports; follow the links, and you will see photographs/figures.
In 1958 we examined a male Sperm Whale 11.6 meters long caught in the Bering Sea, with unusual protrusions in the pelvic region and along the sides of the genital fold (see Figure #1) with a total length of 28 and 34 centimeters, pigmented like the rest of the body. The section of the left protrusion that extended from the body had the appearance of the rounded blade of a propeller, while the right one looked like a fin with finger-shaped processes. The bones were enclosed in dense connective tissue. Judging from an X-ray photograph (see Figure #2) of the skeleton of the protrusions in this specimen, the section of the protrusions that lay inside the body corresponded to the femur, and the middle section to the tibia and the fibula. The section of the protrusions that extended from the body corresponded to the step of the hind limb, and the elements composing it are probably phalanges of the digits. According to the number of phalanges visible in the X-ray photograph, these are the fifth and fourth digits.
This particular whale was a female Humpback of the average length with elementary legs protruding from the body about 4 feet 2 inches, covered with blubber about one-half an inch thick. As shown in the best photograph these legs protruded on either side of the genital opening…
Female Sperm Whale caught by Japanese whaling operation, Nov. 8th, 1956, “protuberances on both sides of the genital opening. … The height of the protuberance was 5.35 centimeters on the right side, 6.56 centimeters on the left side.” (SEE PHOTOS OF THE PROTRUSIONS FROM SIX DIFFERENT ANGLES, Fig. 5)
“Upon examining the interior of the left limb three partially cartilaginous bones were found. They correspond to pelvis, femur, and possibly to tibia, but no joints exist between them. Pretty strong muscles connect between femur and tibia. The tibia is 13 centimeters long for the greater part cartilaginous, and only partly ossified stick-like body with its distal end inserted into the skin of the hind-limb protuberance.
Not being an expert in cetacean biology, I am unable to tell you which regulatory gene(s) behave(s)abnormally to cause these atavisms.
Moreover, consider this additional evidence: in certain species of whales, vestigial leg bones lie unused beneath the skin.
"Nothing can be imagined more useless to the animal than rudiments of hind legs entirely buried beneath the skin of a whale, so that one is inclined to suspect that these structures must admit of some other interpretation. Yet, approaching the inquiry with the most skeptical determination, one cannot help being convinced, as the dissection goes on, that these rudiments [in the Right Whale] really are femur and tibia. The synovial capsule representing the knee-joint was too evident to be overlooked. An acetabular cartilage, synovial cavity, and head of femur, together represent the hip-joint…
You ask why would dolphins have atavistic fins while certain species of whales have atavistic legs… It’s important to remember that there is a lot of variation over time across large populations. It would make sense to me that some cetaceans (dolphin ancestors) would have experienced an evolution toward small hind fins while others (right whale ancestors) did not. Eventually, the hind fins did not provide enough function to justify the hydrodynamic drag, so they disappeared among the population that gave rise to the dolphins.