The definition of a classical black hole is when even electromagnetic radiation can not escape from it. Why then can Hawking radiation be emitted from semi-classical black holes?
What is difference between Hawking radiation and electromagnetic radiation?
First of all, by electromagnetic radiation we basically mean photons. That is particles are radiated by a black hole. Hawking radiation in general can be any type of particles, as in the same way in general radiation can be any type of particles (electrons, etc).
Secondly, the first result you quote 'radiation can not escape a black hole' is a ${ \it classical}$ result (i.e. not that its historical in this sense, but classical in that no quantum mechanics involved) of General Relativity. More generally it is the statement that once inside the Schwarzschild radius (horizon) nothing can escape the black hole. A particles fate is simply moving inwards, eventually ending at the singularity at the center.
Now, Hawking radiation is a ${\it quantum}$ effect. That is, this result supersedes the prior result. Hawking radiation is a consequence of the fact that particle/ anti-particle pairs are constantly being produced and annihilated in the vacuum (that is, the naive vacuum that we think there are no particles in). For such processes near the black hole horizon, sometimes one of these particles falls into the black hole leaving his partner to escape, which has the net effect to an outside observer as radiation from the black hole. In short: Hawking radiation is the more complete description of what happens since it takes quantum mechanics into account.
