The main hazards with traditional nuclear waste is that it is usually liquid sealed inside drums, which fail over time and the waste leaks out. They have come up with ways to solidify the waste through vitrification. They have already built the storage facilities for nuclear waste, vitrification can reduce the space taken up by waste, increasing storage capacity, and massively reduce risks of storage.
Storing it in a stable form makes it available for If someone comes up for a way to use it. Next Gen may be able to do that. It produces electricity at much lower temperatures than needed for traditional nuclear, which is why it could be powered by nuclear waste.