It's a political problem, not a technological problem. We have the ability to use waste to fuel fast-neutron reactors, which not only reduces the amount of waste but makes it less difficult to deal with. The problem is that one output of these reactors would be plutonium, and due to anti-proliferation treaties it opens up a political can of worms.

This solution has been tried for a long time and it didn't work either:

"In 2010 the International Panel on Fissile Materials said "After six decades and the expenditure of the equivalent of tens of billions of dollars, the promise of breeder reactors remains largely unfulfilled and efforts to commercialize them have been steadily cut back in most countries". In Germany, the United Kingdom, and the United States, breeder reactor development programs have been abandoned."

https://en.wikipedia.org/wiki/Breeder_reactor#Discussion

I personally fail to see a solution to the coal waste problem. This gets swept under the rug most of the times, whereas nuclear waste is at least accounted for.

Nuclear waste, according to my understanding, is more of a political problem rather than a technical one. As was the closure of nuclear power plants in Germany.

What is the coal waste problem and how is this comparable to the problem of nuclear waste?

Nuclear waste is a technical problem, otherwise it would have been solved already. There are countries like France where they have 80% of their electricity generated by nuclear power and politics is absolutely pro-nuclear. Yet, they don't have a solution to the nuclear waste problem.