Here is my two cents....
If it is raining light at the start I ride.
If it is raining a lot I sleep or wait to see if it is clear later.
If during the ride it starts raining I ride... I may cut it shorter if I get too soaked or it starts turning colder... Don't risk catching a cold... That would suck.
If I hear thunder or see lightning and I can't ride away/around it I head for shelter. Never ride in a thunderstorm!
Your road bike can take the weather.
Probably the only difference is how you ride...
Be very careful in corners.
Watch out for pavement markers... While and yellow lines, esp new ones, can be very slick. I've seen many people fall just crossing with the line.
Watch out about power strokes. I've had the rear end get loose when I was powering up from a stop light and rolled over a puddle.
Watch out for draw-bridges. Esp ones with metal grates. Those can be very tretourous when wet. Never accelerate on them. Get your speed up to like 10-15mph and then lightly motor across them.
Watch out for train tracks. These too can be esp. slick when wet. Never cross them at any angle other than 90degrees when they are wet (probably even when dry). Also some tracks will have metal plates BETWEEN the rails and these are just DANGEROUS. Be extremely careful crossing them.
ALso some road construction will use large metal plates to hide work-pits on the road. These too are very dangerous in the rain/when wet.
The reason to be more careful about the above, not the bike, is that you will probably be going faster than you would in the mud with your ATB and also if you fall you will fall hard on the pavement or under a moving car.
Other than that... Have fun in the rain!!! :p
JBP