This is a deep and often loaded question. If you are agonizing over the decision to go with either a carbon or aluminum rim, here are some pro and cons that should be helpful.

Carbon rims are typically very durable, however they do not generally respond well to extremely sharp impacts (such as potholes). While most carbon rims can handle relatively harsh environments pretty well, the cost of replacement is significantly higher than aluminum.

Aluminum rims will often bend when subjected to sharp impacts. This may require the rim to be replaced, however it can typically be ridden (cautiously) until it can be rebuilt. 

Carbon rims will rarely fail catastrophically. Older (pre 2011) carbon rims and rims from suspect manufactures can sustain damage through excessive braking heat. 

Aluminum rims rarely fail under braking.

Carbon rims will generally outperform aluminum rims in the wind tunnel due to their aerodynamically efficient cross-sectional designs. 

Deeper section rims can be significantly more difficult to handle in strong cross wind conditions, so don't always assume that the deepest possible wheel will be your best choice for all riding conditions. Improving aerodynamic performance will almost always deliver better results than reducing weight. There are exceptions to this rule, such as hill-climb specific wheels. 

If you plan to use your wheels for race-day use only, the carbon wheels make a lot of sense since they will probably be used for far fewer daily miles on tough city roads.

If you use your wheels for both training and racing, be sure to select a carbon or aluminum wheel with the appropriate spoke count to handle the additional stress of daily riding. Most off-the-shelf carbon wheels have minimal spoke counts that may not be appropriate for daily riding. 

Aluminum braking surfaces generally stop in all weather conditions and do not require special brake pads. Carbon braking surfaces require special brake pads, do not stop as well in wet weather, and are more sensitive to overheating under heavy braking conditions. 

Beauty is in the eye of the... , however we all agree those carbon wheels look cool on your bike. 

Carbon rims can be a nice addition to your ride for both training and racing use, however they will generally cost more than aluminum and require a bit more care during use. If you need significant aerodynamic improvements, or just admire the aesthetics of your wheels, carbon wheels can deliver the goods. Aluminum wheels generally offer excellent ride comfort, solid cornering performance, and ease of repair at a lower price point than carbon. Aluminum wheels make a sense if you are looking for a single set of wheels that can be used in all weather and road conditions for training or racing.