Why should her father cater to her every whim?
He has always indulged her every whim.
They seem to be synonymous and interchangeable. But can we say that in everyday English one is more common than the other?
'Cater to' may often be neutral and means "supply a desire, need or requirement", e.g. the Acme Screw and Bolt Company caters to the needs of industrial clients for threaded fasteners, but 'indulge' can have the pejorative implication that the receiver is greedy or demanding and the supplier is weak or easily persuaded. However, indulgence can be acceptable or even praiseworthy - my four year old niece is fond of ice cream and when I took her to the zoo last week I was happy to indulge that. She also claims that her teddy bear can speak, and has decided opinions about chocolate, and I was happy to indulge her in that belief.
Cater to is neutral: it expresses no judgement about the appropriateness of the behaviour.
Indulge is not neutral: it implies that the demands are unreasonable, or that the person satisfying the demands is behaving unwisely.