Both describe something undesirable continues all the way, but Vする一方だ implies escalation, and Vするばかりだ does not. In this sense, you can replace Vする一方だ with （ますます／どんどん）Vするばかりだ.
If the verb already has a cumulative meaning, then the two expressions make no difference.
死傷者は増える一方だ。 = 死傷者は増えるばかりだ。
Casualties keep on increasing.
Otherwise, these two would not be interchangeable.
子どもはおびえる一方だ。 The child only gets more and more frightened.
子どもはおびえるばかりだ。 The child can do nothing but being frightened.