It might be a bit too late, but I noticed there is no answer to this question, so I thought I'd give one.
In the scanning electron microscope (SEM) image above, the carbon nanotubes (CNTs) are the thin hair-like entities; so you might as well say the "fur" is the nanotubes. But first, maybe some context.
CNTs can be thought as folded sheets of graphene, therefore they share many similar characteristics and properties. They can be single-walled or multi-walled (nested nanotubes, equivalent to folded multi-layer graphene), with diameters typically 1-2 nm and 5-15 nm, respectively. CNTs are grown with a variety of ways, with chemical vapour deposition (CVD) probably being the most popular one, especially in research. Nanotube growth with CVD is a bit random in the sense that there is no direct control of the growth length or direction, but the growth density can be controlled. In the SEM image above, you can say the growth density is quite high since the CNTs are all over the place.
But if nanotubes are typically a few nm in diameter, why do they appear so much wider in the SEM image? This relates to how the SEM works and how electrons interact with materials of different conductivity. When imaged properly, CNTs on insulating surfaces have apparent diameters that are about 50 times larger than their true diameter. Moreover, nanotubes on the background appear darker than those at the "front". This again depends on how the electrons from the SEM interact with the material they are imaging and how many of those electrons make it back to the SEM detector. This paper goes into it in more details.
In general, it is very unlikely for the hexagonal structure to be visible with an SEM. However, a scanning tunneling microscope (STM) could do the job: you can see a CNT image taken with an STM here, where the hexagonal structure is visible.
Because diagrams lie. They have to, in some cases, to make their point clear.
I presume you mean something like this?
(Courtesy "Simetrical", cc-by-sa 3.0. Wikimedia Commons. Retrieved from: 1)
Yes, the tube shown in the picture is artificially straight. It's no different from showing a diagram of, say, an electric wire that shows it as straight. It's meant to illustrate the structure, not the physical properties of it as an object, such as its deformability. Just as a piece of cable can bend, so too a nanotube can bend. They're flexible. The flexibility is because the bonds are not perfectly rigid, but can strain (deviate from their customary angles).
Also, you cannot see the hexagonal structure because it is too small. The width of a nanotube (not illustrated in the picture I give) is like maybe only 2 nm or so; and the length of a single carbonic bond is something like 0.14 nm. Note that the scale bar on your picture translates to 2000 nm, so the nanotube "hairs" are probably on the order of 1000 nm long. The bonds and atoms are thus just far too small to see in the picture. Equivalently, it's like you're "zoomed way out" compared to the level of detail in a diagram like the one above.