Aluminum or Carbon...which is stiffer. The answer really depends on how you manipulate the material. You can use either material to build a "flexy" flyer or the stiffest bike you'd ever want to ride. It's all in how you design the tubes (wall thicknesses, tube diameter, etc) and the frame geometry.

I would want to answer this question in terms of advantages/disadvantages of each material.

The advantages of aluminum are its low costs with respect to strength to weight ratio. It is also relatively easy to form this material into different shapes and it is not difficult to weld. One disadvantage is that of all the frame materials used (steel, aluminum, titanium and carbon fiber) it will fatigue quicker (i.e. fail)due to the nature of the metal. I would consider this a factor if I were choosing a mountain bike frame that would be heavily thrashed for many years. For a road bike, it's a non-issue. If it were an issue, jets would be built of something else.

The advantages of carbon fiber are that of all the frame materials used (steel, aluminum, titanium and carbon fiber) it is the strongest (based on engineering tables that calculate relative strength) of all of them which means that correctly designed, you will be able to build a frame that will be strong, stiff and lighter than frames made of any other material. Another advantage of carbon fiber is its ability to damp vibration. This means that you will feel less road shock. I have seen occur often however, where riders are convinced that aluminum frames are stiffer than carbon fiber because with the carbon fiber, you don't feel the harshness. Some riders think a bike has to ride harsh to be stiff...this is nonsense. If many of the top sprinters in the TDF are using carbon fiber, the issue of adequate stiffness has been answered.

The big disadvantage of carbon fiber is cost. While not an issue for pro riders, the purchase of a carbon fiber frame may be beyond the budget of some riders.

A note on titanium: of all the materials, this one is the most difficult to shape which is one reason you don't see a titanium frame with the subtle curves of a Trek Madrone. You could do it, but the cost of shaping the tubes would be astronomical. Carbon fiber has leapfrogged titanium as the material of choice for "cutting edge" frames because you can build a better performing frame that in the end will cost less to produce. If it were not so, you would see top pros riding ti, not carbon and the top companies producing ti, not carbon fiber bikes for their hi-tech offerings. It is the same with other sports. Carbon fiber, not ti
is used for applications on Formula One car bodies because you can shape it!

The bottom line: Establish a budget, then ride a bunch of bikes. Personally, I would not hesitate to buy a bike made of any of the materials mentioned if I liked it the best. All of them, if built by a reputable company will be more than adequately strong and will be backed by an excellent warranty.

When you ride, try some of the aluminum frames with carbon fiber stays and fork. These cost less and you get the benefits of vibration damping with the carbon fiber used in these areas.

Happy hunting!