Does it take longer to generate an address when I stop and restart vanitygen? I want to generate a bitcoin address.
 I start vanitygen, but a power loss occurs.
 Now I restart vanitygen.
How does this affect the time to generate the address?
tl;dr If you stop vanitygen for 5 minutes, on average you've added 5 minutes to the time it will take to complete.

While it is running, vanitygen tells you how much longer it will take until you have a 50% chance of finding a suitable address.  About half the time, that countdown will reach zero, and then the countdown starts again, but this time showing the time until you have a 75% chance.  And so on.

This gives the impression that it is making "progress" towards finding a suitable address, but that is misleading.  Every second that vanitygen is running is has a certain chance of finding an address.  There's no progress being made, in the same way that you don't "make progress" towards winning the lottery if you play it every week.  Just eventually if you play enough times, you win.

So if you stop the program running and start it again, you've lost nothing other than that you've wasted the time between stopping and restarting when it could have found a solution.