Well martial arts derive from 'arts of war' i.e. formalised and diligent training methods to condition the soldier/warrior and to impart techniques which are useful in battle.
A self-defence is a way of defending oneself.
With less battles, both in the east and in the west, many of the techniques originally from martial arts have been incorporated into self-defence.
'Modern' martial arts aren't really conducted with the aim of war in mind (maybe the best equivalent to martial arts now would be basic training in the army). However it seems they have retained the name as they are taught in this formal quasi-military way, not just to impart self-defence techniques but to increase athletic ability/ foritude/ courage/ honour etc.
Thus I think the difference is mainly that modern martial arts retain some of their historic roots and aims, and self-defence doesn't. (but obviously there is a cross-over).