Putin might be mad, but they wouldn't give 2 fucks about Snowden, if Russia got anything in return for granting him asylum they already got it.
That said the US would more likely want to exfiltrate him for this specific reason, they would want to know if and what did Snowden handed over to Russian intelligence.
https://en.wikipedia.org/wiki/Poisoning_of_Alexander_Litvine...
Putin might be mad, but they wouldn't give 2 fucks about Snowden, if Russia got anything in return for granting him asylum they already got it.
That said the US would more likely want to exfiltrate him for this specific reason, they would want to know if and what did Snowden handed over to Russian intelligence.