Your mileage with Cyanogen Mod (CM) depends on the phone you are running. If you have an HTC phone, you will not have any HTC Sense running or installed at all. So HTC widgets will not work. This is a hurdle for many HTC phone users that enjoy some of those widgets. Personally I've been able to find adequate replacements to them. So that wasn't a barrier for me.
If you are not using an HTC phone there probably won't be a lot of cons. Again though you might not be able to use any stock widgets that came on your phone. But you might have a different dialer and contact app then what you are used to.
Do you connect to an email system at work, such as an Exchange mail system? I do and that was one of the reasons for me not using CM. Not that it didn't work, but there was some functionality I was missing, a few settings and I didn't like the way the contact app managed the Exchange contacts. Whatever HTC uses for their builds worked better IMO so I use an HTC based ROM.
As Saboogu states, hardware will make a difference too, but CM team is good about refining their ROM and working out kinks so that it functions well even on older phones.
If you are already rooted make a NAND backup first. Follow this guide I wrote keeping in mind it was written for an Evo, but most info still applies to other phones. Once you have a backup then wipe and flash and go from there. See if you like it and the feature set. This way if you hate it you can easily restore your backup.
The biggest goal of CyanogenMod, in my eyes, is to provide back porting of functionality to older phones who are not getting the latest version of the Android OS (2.1+), e.g. the G1. It also gives plenty of little tweaks here and there to add some functionality to your phone that you may not have gotten before.
It also (typically) requires rooting your phone, so you also get those benefits too.
Although CM releases "stable" versions, your stability may vary. You'll most likely see more reboots from CM than a stock rom or other miscellaneous bugs.
Really depends on your hardware. At this stage, CM6 is likely slower on a G1 than the OTA 1.6 release. Maybe not quite as stable either, but certainly more feature-packed.
On the other hand, on a higher memory device that has native 2.x support, those same drawbacks won't apply.
I previously owned a HTC Hero and I resorted to CyanogenMod (CM) because it does not contain HTC Sense.
However due to hardware limitation I noticed that my phone ran slower as the Android version increase. Since Froyo the phone no longer fast and there are times it lags.
Another problem I faced was hardware support. For example camera may lack of some features introduced by HTC.
One drawback is reduced support for hardware in some cases. IE Cyanogenmod for HTC Desire doesn't support HD video.
We have an Orange San Francisco, AKA ZTE Blade in the family.
I would say the biggest weakness I found with the CM7 rom for this device is around various little bugs with proximity sensor, wifi connection and power management states.
IMHO this is due to ZTE being very hacky in the way they maintain kernel source, and partly due to having undocumented hardware revisions which causes various minor glitches. This caused extra work for the devs who put the Blade kernel together for CyanogenMod, but all issues have been fixed.
Sometimes users suffer bugs and simply do not report the bugs on the bug tracker so the devs are unaware of issues to fix.
All these are minor issues, against the well know pros of CyanogenMod. I have had far more stability, performance and support from CyanogenMod than I did from Orange and their stock ROMs.
