If aerographite is lighter than air, why doesn't it float? Air is 6 times denser than aerographite but looking at pictures or videos presenting the material, I see it resting on tables rather than raising to the ceiling.
Also, since the material is made of carbon nanotubes, I assume there are empty spaces between those tubes. Why are those spaces not occupied by air?
 A: The density measurement was done in air, not vacuum.
On page 24 of the Supporting Materials document it states:

Determination of masses and densities:
The dimensions of Aerographite specimens had been determined by light microscopy. Mass of samples had been
  measured on standard atmospheric conditions with a Saratorius MC5 microbalance (Capacity / Readability: 5.1 g x 1 μg; Repeatability: ± 1 micrograms).

(emphasis added)
This means that the weight (and therefore the density) did not take into account the buoyancy effect of the surrounding air. The actually density is the published value PLUS the density of air. 
A: Your two questions are connected. There is a huge amount of empty space in aerographene (and other aerogels). However this space is filled with air, and precisely because it is filled with air it doesn't float. 
This is because the density reported is the density the material would have if the air was sucked out (i.e. in vacuum), and it is so low because the material is extremely porous. But in the atmosphere, the air fills the immense empty space. The effective volume of air displaced by aerographite now takes up only the volume of the constituent nanotubes of aerographite, which is extremely small. The tiny weight of this displaced air presents the buoyant force, which is not sufficient to counter the weight of the structure. Effectively because it is so porous the aerographite's density increases when not in vacuum.
On the other hand, given that graphene is known to be non-permeable for atoms, if you sucked the air out of aerographene and encased it in graphene, and if outside air didn't squish the whole thing so that it's density surpassed that of air, than the resulting balloon might float.
A: What follows is very much the same as mgphys' answer, but I'm going to be pedantic about what I mean.
So imagine that 


*

*I manufacture a substantial body of areographene, and then carefully slice out of it a rectangular prism that is $h$ by $l$ by $w$ in size. This gives a volume for the material of $V = h l w$.

*A put a analytic balance in a vacuum chamber and careful measure the mass $M$ of the test sample. Now I calculate a figure for the bulk density $\rho_\mathrm{bulk} = M/V$ which will be rather less than the density of air. This is the kind of measurement that people mean when they say that areogels and areographene are less dense than air.

*However, if I retrieve my sample and look at it under a microscope I will see that is has a fine scaled structure in which filaments and sheets of graphene do not fill the volume of the material but instead form an open lattice work. So the real volume of graphene is not $V$ at all!

*If I choose a liquid that will not damage the material I could carefully measure the volume $v$ actually occupied by the graphene using Archimedes' method.

*With that I calculate $\rho_\mathrm{detail} = M/v$ which will be higher than the density of air. Now, because the air can move into the spaces between the filaments and sheets the volume of air displaced by the sample is $v$, not $V$ and the density we have to use to determine if it will float is $\rho_\mathrm{detail}$.
This is also why mgphys states that if we were to wrap up the sample with a thin, non-permeable membrane while in vacuum (and that membrane held, and the sample doesn't get crushed) it could float.
