This is true, the waste issue is different with these two technologies, but I don't think it's all that significant in either case.

Fission produces some awful waste, but what I like to point out is just how little it produces. My favorite example is nuclear submarines. Nuke subs have to come to port every so often for food, equipment, supplies, etc, but not because they're low on fuel. They don't carry a lot, about 500kg (half ton) and that lasts them a very long time. So how often do they need to be refueled? Once, most subs are refueled just once in their ~30 year lifetime. Some subs will be decommissioned before ever refuelling, using just one set of uranium fuel rods for their whole life.

Edit: I wanted to visualize how much 500kg is, and I know uranium is heavy but I really didn't have any idea what a half ton would look like. Turns out, it's about 26 liters, 1 cubic foot. (Though, ideally your uranium wouldn't be measured in either of those units, you really don't want liters of liquid uranium, and that's exactly where a solid cube is headed too...)

Given the tiny volume of waste produced over such a long time... We can figure out the storage. Even if the solution is costly, there's really not much to store, this is very manageable.

So yeah, I'm not saying waste isn't an issue for nuclear power, it is. But I think it's not the biggest drawback, it seems like the overall cost is still the bigger problem in operating a plant.