Uranium 235 can decay by splitting into two smaller nuclei and releasing some free neutrons. Frequently those smaller nuclei are themselves radioactive. Uranium has a very long half life, on the order of billions of years, so the natural rate of decay is very low. This makes it relatively safe, since there is not too much radiation produced per second. Many of the product nuclei that can be formed have a much shorter half life. So they emit more radiation, though on the plus side they don't last as long. In nature, these smaller nuclei are still formed, but at a very low rate. In a nuclear reactor, the uranium is bombarded with a large number of neutrons, and it has a chance of splitting whenever a neutron hits it. So there's a high rate of decay, and as a result, once the fuel has been in the reactor for a while, it contains a large amount of those fission products. That's what makes the waste more dangerous: it has isotopes in it that decay faster and therefore emit more radiation per second than pure uranium.

This is a really helpful explanation. I didn't have a clue that the long half life actually results in lower levels of radiation.

You can hold natural uranium in your hand for a long time before it becomes a health concern. It's toxicity (heavy metal!) is more dangerous than its radioactivity. You shouldn't try the same with spent fuel or you will lose you arm quickly.