Nimoy's voice changed because he got old, just like his face and hair color changed. It happens to all of us eventually, if we're lucky.
TMP was only ten years after the third season of TOS, which is not a huge amount of time, not a quantity of time that should result in large vocal changes. Mr. Nimoy was only 48 at the time of TMP, which is not old at all. Sure, his voice sounds old in the reboot movies, but he was 78 by 2009. There's a big difference between 48 and 78.
Ten years can make a world of difference, especially as you approach middle age. The years start to take even more of a toll after 50. Check out the changes in Shatner's voice from TOS through the movies, on to Boston Legal and today.
The rest is ... wait for it ... acting