A road bike should be fine for the commute. I've commuted on one for years. A cyclocross bike really won't offer anything that a road bike doesn't offer. You can find both with plenty of clearance for fenders and with good brakes. The main difference between the two is cable routing and tire tread. You don't need all your cables routed on your top tube for a commute and you would probably be better served with smooth tires to give you better grip on city streets, so a road bike makes more sense in those respects.

Do you have any super steep descents (I.e. 20%+) on your commute? If so, you might appreciate disc brakes, which are a little easier to find on a CX bike.

Overall, I would recommend a road bike since you'll appreciate it more on the long weekend rides, and it has no downsides for your commute as long as you stay on paved terrain.