to me, they both express politeness but "kindly" shows more friendly feelings. "please" is a bit asking or requesting in a most polite way. more often than not, "please" is used based on the given rules, conduct or etiquettes. In a library for instance, one significant rule is maintaining the silence so you will notice sign boards on the wall saying:
"Please Keep Silence" (and not 'Kindly Keep Silence').
on the other hand, when we are using "kindly", we are establishing a friendly conversation with someone. It is important to know that someone who will do a favor to you, will do it not just because they are obliged to obey you, but they are happily doing it.
:)