Who throws radioactive waste out at sea? High level waste (fission products recovered from spent fuel) tends to be stored dissolved in nitric acid in cooled stainless steel tanks. There are liquid effluent discharges but these tend to make up a small minority of the total radioactive inventory from spent fuel.
Shooting stuff in to space always has the associated hazard of possibly ending up back spread all over the earth due to a rocket failure - not the kind of environmental hazard even BP would contemplate at the moment!
Actually, while you will often hear the "launch vehicle could explode and then radioactivity would be dispersed everywhere" argument, I don't think it's actually a valid argument.
Look at the space-based RTGs we've been using ever since pretty much the beginning of space exploration. They have the radioactive material packaged up in a container which will survive the most extreme conditions, including catastrophic failure of the launch vehicle and even atmospheric reentry without releasing the radioactivity.
So clearly if we wanted to shoot radioactive waste into space it is possible to do so with no danger.
The real reasons are simply that it would be incredibly expensive and it's just not necessary.
Radioactive waste is not a huge drama or problem like some people imply it to be.
There's not that much of it, it's being stored safely, most of it is useful material, and it hasn't ever hurt anyone.