Explain why the gem-dimethyl groups appear as separate peaks in the proton NMR spectrum of isoborneol while they are an unresolved singlet in borneol.
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The gem-dimethyls in borneol (the molecule on the left in the picture) are in similar chemical environments. It doesn't really matter that the OH group (never mind that it is shown here as a doubled bonded oxygen) is a little closer to the methyl one methyl than the other. Both dimethyls 'see' pretty much the same chemical environment.
In isoborneol, the OH group is in the axial position and sticks up above the plane. It is now much closer to one of the dimethyls than the other. The dimethyls 'see' different chemical environments - that is, the other parts of the molecule affect them in different ways.
So instead of getting one unresolved singlet (one peak) as in borneol, you see a split peak in isoborneol.