The show has been pretty sloppy with the scientific aspects of how radiation and mutations work, so there's not really any point in trying to explain what's going on using real-world science. (It would be off-topic here anyway.) Mostly they tend to treat radiation damage as a disease you can build immunity to (up to and including the bone marrow transplants "healing" it).
In this sense, the difference between the ground radiation and solar radiation would be in dosage. (This is actually broadly accurate, though the details are a bit off.) The people on the Ark would be receiving doses of high-energy solar radiation constantly, but at such a low level that the damaged cells would be destroyed and replaced before they could cause harm. (If this wasn't true, then astronauts would all come back to Earth teeming with cancer cells.)
The ground radiation, on the other hand, is a significantly higher dose, and the people and wildlife on the surface is literally walking through it all the time. Soon after the nuclear event, there would have been tons of it, and as we've seen, most things that were alive then are dead. What survived were those creatures that the radiation damage to their DNA wasn't lethal. This is another case where the show is broadly correct: major genetic mutations would likely show up in offspring moreso than the adults that were exposed: their bodies are already fully formed, and any deviation in their DNA would, at best, cause cancer, and at worst cause organ failure and death. However, it's possible that a mutated embryo might survive, assuming the mutation was non-lethal.
As time passes the radiation levels are dropping. It's low enough that the Ark and Grounder populations "immunity" is able to cope, though it's obviously still strong enough to kill the Mountain population in minutes. Thus, the immune populations are mostly healthy adults having healthy children, but there's still a chance of radiation damage to the reproductive system leading to a mutated child.
The as-yet-unanswered question is whether the "immunity" the Ark people have will prevent their children from also, occasionally, being mutated. In theory they should have about the same rate of abnormal birth as anyone else on the ground, but obviously it's still to early to know that.