Both of these expressions are, of course, figurative. Kids can't literally drive a man up a wall, and, although there is evidence that prolonged stress can cause premature graying, there is no way for someone to be aware in the short term that the process is occurring.
There is considerable overlap between the two expressions, in that both of them indicate that the kids are causing the speaker a great amount of stress. My general sense is that (1) conveys the sense that the stress takes the form of severe irritation and that its effects are immediate and intense, whereas (2) may imply that the stress is more due to worry, and that its effects are more prolonged and far-reaching.
But this is one man's opinion. Both expressions are informal and not to be taken literally, so they are subject to a rather broad range of interpretation.