OK, then do you have some kind of 'stay-alive' (keepalive) set with puTTY?
They both "seem" to run fine if they're busy, you just need to look at whether the hosting server kicks you out. This may have nothing to do with macTerminal vs puTTY on Win...I'm thinking out loud here.
But, FWIW, my puTTY sessions do abend frequently in XP or Win2000, on a real PC (not parallels). 5-10 minutes getting kicked out of a Linux server after inactivity doesn't sound unusual, I guess.