This is a major political power maneuver. As long as Snowden is in Russia, Obama can control the messaging. He is, after all, cooperating with an important physical threat to, if not outright enemy of the United States. Asylum in Switzerland, along with all the mental associations with its neutrality and sovereignty, puts Obama in a much trickier position. Probably not enough to turn the tide, but it forces Obama to address the situation with a subset of his constituents who are not as easily convinced by "Murica is Awesome!" style arguments.
Snowden leaving the country would provide good timing for Russia to de-escalate their Western relations. On the other hand this could be years out. I suspect Putin may rather cut off the gas while waiting for another major economic contraction and debt default cycle breaking up at least pieces of the EU.
Switzerland now has grudges against the US for destroying their bank secrecy. If Bloomberg news is to be believed, both capital and humans are leaving the country as a result.
