Water – tar joint between garage wall and asphalt

Depending on the size of the cracks an asphalt sealant should close things up and protect the driveway. If the cracks are large you might have to get some asphalt patch and then seal it. your local home improvement store should have asphalt sealant and may have asphalt patch available.

Asphalt sealant is pretty messy stuff though, best to wear throw away clothes and boots.

First of all, what you describe sounds like job done improperly. Perhaps you can even use the contractor warranty and have it fixed for free.

Asphalt pieces put from different loads will usually have perfect adhesion to each other - the previously put asphalt is still very clean when the newer asphalt is put, so new one will attach to it perfectly as if it was a single piece with almost no joints. Since you see asphalt peeling off there's likely a problem - most likely the asphalt or the base underneath it got dirty and that prevented good adhesion. No good adhesion - prepare for more peeling off as time progresses, especially since you have temperature variations and freezing water.

IMO it's worth cutting asphalt at the flaw spot deep enough and sealing it with a cold patch to prevent further destruction. If destruction progresses along the joint the same should be done along the whole joint.