For what bioinformatics tasks is Biopython more adapted than Bioperl? Are there any advantages to learning Biopython instead of learning Bioperl?

Ideally, we would learn both, but someone starting out in bioinformatics may have to choose what to learn first depending on the kind of problems actually encountered.

Are there problems for which Biopython is better than Bioperl (or vice-versa)?
Regading the perl vs python discussion, there is no final answer which language is better, but I have some advice for you:

Learn the language your colleagues or your advisor use. This way you are able to discuss your code with them and also get help if you run into problems.