GR does apply to both cases, but you get different results. Space-time itself can carry energy/momentum; in the big bang, this energy/momentum helps drive material outward in an expansion of space. That's not present in a standard black-hole.
That is good logic, but it misses some of the details of inflation and the big bang---you can think of it in terms of there just being a really strong expanding energy/momentum during the BB which was great enough to resist the gravitational pull of so much matter.
Unfortunately, that's part of why the balloon surface is only an analogy. In GR space-time and matter are coupled together, and are coupled in a highly non-linear (and thus complex) manner. Let's try a different analogy:
Lets say you throw some object (maybe a stone, maybe a model boat, whatever) across the surface of a pond. That object will cause ripples in the surface of the water. Those ripples and waves themselves could cause the object to move---and then the new motion of object could cause further waves, motions, turbulence etc in the water.
In the case of GR the interactions are far more complex and long-ranged.