Re: Romex has 3 wires but new timer switch also has a red wire
I agree with Jack.
It appears that you have what's known as a "switch loop" -- one insulated wire is is the hot supply line, the other is the feed from the switch to the fan. The white wire is not a neutral, and it should be colored with a black marker or electrical tape to indicate that it isn't neutral. Marking is now required; when your home was built it may not have been required, or maybe the electrician was lazy.
Your switch box apparently doesn't have a neutral wire. The timer switch you selected requires a neutral wire to work properly. As Jack said, you will need to use a mechanical timer, or replace the wiring to the switch location so that you have a neutral.
A recent change to the codes (2011 National Electrical Code) now requires a neutral wire in every switch box for this very reason: to support electronic timers & occupancy sensing switches. However, previous wiring is grandfathered, so there is no requirement to retrofit except during remodeling.
Note that all of the above is based only on what you told us and serious speculation on my part. I can't see your exact situation from here, and there may be important information that I'm missing.
The "Senior Member" designation under my name doesn't mean I know a lot, it just means I talk a lot.I've been a DIYer since I was 12 (thanks, Dad!). I have read several books on various home improvement topics. I do not have any current code books I can refer to. I was an apprentice plumber for two years.