Reconciling the two aspects: carbon is stiffer for power, but offers a softer ride

Depending on how the carbon fibres are laid up, the flexibility can be altered by a huge amount. Different fibres such as kevlar and glass can be added to the mixture to increase flexibility in areas and specific directions where it is wanted.

In this way a carbon frame can be both stiffer and more compliant, but in different areas of the frame depending on where the designer would like these attributes. It is not possible to do this to such an extent with other materials.

Would you like some examples? The GT grade in carbon was an interesting bike on launch with 4cm ''virtual suspension'' travel in the rear yet an excellent power transfer.

http://s3.amazonaws.com/gtvendors/gorogue/img/content-pic-triple-triangle.png

From the website:

The solid core provides incredible rigidity, and the carbon outer layer acts as added vibration dampening. In the end, you have a bike that gives the rider a lot more control and a lot less fatigue on the ride.

Bianchi use kevlar fibre for the Infinito CV seat stays to get a similar if less dramatic effet.

A simple explanation I've heard is that metal tubes are isotropic - their material properties are identical in all directions (tube shape and thickness being constant). Carbon fiber can be made fundamentally anisotropic, i.e. their stiffness and strength differ in various directions.

I am not an engineer, but this brief synopsis was given by Josh Poertner on his Marginal Gains podcast. He has extensive engineering experience in the bicycle industry.

I also suspect he was simplifying for a non-engineering audience. For one, we are not limited to round and only round metal tubes of equal thickness throughout. For quite a long time now, we have had butted metal tubes, i.e. the tubes are thicker at the ends where you do need more strength because you're welding that tube to others, but they're thinner towards the tube center; for spokes the butting is external, but for frame tubes it's internal so you can't see from outside. For another, you can shape the tubes. For example, I know that at least some metal down tubes are oval where they join the bottom bracket - the greater lateral cross section makes for a stiffer bike. More complex shaping than that is available, as the comment below alludes to. Within material limits, you can change the diameter of the tubes as well, where larger and thinner walled tubes are stiffer than smaller and thicker ones. Somewhat related to that, variations in metal alloys can enable slightly different material properties. For example, 6/4 vs 3/2.5 titanium, steel with niobium in the alloy mix may have enabled very thin-walled (and thus large diameter) tubes that are stiffer, various alloys of aluminum. So, metal may be isotropic if you hold the tube dimensions constant, but there are still numerous ways to adjust the frame properties of metal bikes.

For another, it may be more correct to say that most carbon fiber structures can be quasi-isotropic, although you can also make them close to anisotropic - it might be more technically correct to talk about degree of isotropy. You can, I believe, make an isotropic carbon structure if desired.

Regardless of the specifics, it appears you can vary the material properties of a carbon structure more, and I believe much more, than a metal structure of the same shape. With carbon, you're not limited to changing the shape and wall thickness of the tubes involved. There are more degrees of freedom to vary the properties in.

One last bit as to the notion that carbon offers a more compliant ride. It is true that carbon can damp vibrations. However, a lot of your vibration damping comes from your tires. Even performance road bikes these days can take relatively large tires. You can safely run larger tires at significantly lower pressures than we all used to, and you are likely to have less rolling resistance, not greater (this is actually a complex topic, and that statement is an oversimplification). Also, steel and titanium have long been regarded as comfortable rides; they may feel qualitatively different than a carbon frame, but none of those materials may be clearly superior in overall subjective comfort. For that matter, aluminum frames have improved considerably since the 2000s, even though that material has been overshadowed by carbon. It's possible to make a comfortable aluminum bike as well. It is not pure marketing to state that carbon can be made not isotropic, so a carbon fork can be more compliant vertically (i.e. it damps vibrations) than it is laterally (i.e. it doesn't flex side to side when you pedal). It is not marketing to state that carbon bikes can be designed that way as well.

For fork material, it does appear that carbon has come to dominate. Steel forks can also be comfortable, but they do have a weight penalty. While the amount of weight in question actually only has a minor impact on performance in most cases, it is a pretty visible parameter, and anyway steel forks don't seem to be a mass production thing for performance bikes. (FWIW, I have a custom steel bike with a steel fork.) I'm pretty sure aluminum forks were always harsh. I know that titanium forks never made headway, in part due to the cost, and possibly because some of the earlier ones had failures. I'm not sure if that was a quality control issue or something inherent to material properties.

How can these two aspects be reconciled?

Easy! It's marketing. Marketing need not be true. It only needs to be something a stupid buyer will believe.

Even a high-pressure narrow road tire has many centimeters of flexing. Compare that to the flexing of the frame or fork, probably measured in millimeters, you see that the flexing of a frame is nowhere near the flexing of a tire. If you buy a frame or fork that supposedly offers a "harsh" ride, and put 32mm tires on, and compare that to a frame or fork that supposedly offers a "soft" ride, and put 23mm tires on, you'll see the bike with 32mm tires is more comfortable -- assuming you didn't put equal pressure in the tires but rather adjusted the pressure appropriately according to the tire width.

The only design criteria of a frame is that it must be stiff. It is achieved in the rear by using a tetrahedron comprised of the rear wheel axle, two seat stays, two chainstays and the seat tube. In the front, it is achieved by using a triangle comprised of the seat tube, the top tube and the down tube. Because a tetrahedron is a three-dimensional structure offering stiffness in all directions, but a triangle is a two-dimensional structure lacking stiffness in the third dimension, the triangle has large diameter tubes, far larger than those used in seat stays or chainstays.

About the only location in a bicycle that can flex somewhat in addition to the tires is the saddle. The reason is that the saddle has far thinner rails than the diameter of the frame tubes, so the rails flex, and also the plastic in the saddle can flex too.

Frames and forks however can affect the feel of the ride, not through flexing, but through their geometry. For example longer chainstays offer a comfortable and stable ride (unfortunately the trend today is that long chainstays are not fashionable whereas chainstays so short that the seat tube must be specifically shaped to make room for a 23mm tire behind it are fashionable -- and don't try fitting a 25mm tire in there!). The head tube angle and fork rake also affect ride feel.