Cisplatin is an ancient compound that was FDA approved in the 70s for certain cancers. It is a simple drug PtCl2(NH3)2 that acts by damaging DNA. It does have a number of side effects that make it for some people problematic. Ringing and hearing probles are among these problems.
Carboplating is a newer drug that very similar to cisplatin. It was introduced in the late 80s. It has less side effects than cisplatin. It is also less reactive but can be given in higher concentrations. It does cause amongst other things a drop in the white blood cells. This is (or should be monitored) when you are on this drug.
From what I have seen MOs generally use cisplatin because it has been around longer, there is more data and it is though to be more potent that carboplatin (that may however depend on the type of cancer). Quite often people start out with cisplatin and then get switched to carboplatin .... as happened in my case.

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