Why is BCl3 a monomer whereas AlCl3 exists as a dimer?
Introduction

The bonding situation in $\ce{(AlCl3)2}$ and $\ce{(BCl3)2}$ is nothing trivial and the reason why aluminium chloride forms dimers, while boron trichloride does not, cannot only be attributed to size.

In order to understand this phenomenon we need to look at both, the monomers and the dimers, and compare them to each other.
Understanding the respective bonding situation of the monomers, is key to understand which deficiencies lead to dimerisations.

Computational details

Since I was unable to find any compelling literature on the subject, I ran some calculations of my own. I used the DF-M06L/def2-TZVPP for geometry optimisations. Each structure has been optimised to a local minimum in their respective symmetry restrictions, i.e. $D_\mathrm{3h}$ for the monomers and $C_\mathrm{2v}$ for the dimers.
Analyses with the Natural Bond Orbital model (NBO6 program) and the Quantum Theory of Atoms in Molecules (QTAIM, MultiWFN) have been run on single point energy calculations at the M06/def2-QZVPP//DF-M06-L/def2-TZVPP level of theory.
A rudimentary energy decomposition analysis has been done on that level, too.

Energy decomposition analysis

The dissociation energy of the dimers $\ce{(XY3)2}$ to the monomers $\ce{XY3}$ is defined as the difference of the energy of the dimer $E_\mathrm{opt}[\ce{(XY3)2}]$ and double the energy of the monomer $E_\mathrm{opt}[\ce{XY3}]$ at their optimised (relaxed) geometries $\eqref{e-diss-def}$.
The interaction energy is defined as the difference of energy of the relaxed dimer and double the energy of the monomers in the geometry of the dimer $E_\mathrm{frag}[\ce{(XY3)^{\neq}}]$ $\eqref{e-int-def}$. That basically means breaking the molecule in two parts, but keeping these fragments in the same geometry.
The deformation energy (or preparation energy) is defined as the difference of the energy of the optimised and the non-optimised monomer $\eqref{e-def-def}$. This is the energy required to distort the monomer (in its ground state) to the configuration it will have in the dimer.
$$\begin{align}
E_\mathrm{diss} &amp;= 
  E_\mathrm{opt}[\ce{(XY3)2}] - 2E_\mathrm{opt}[\ce{XY3}]
  \tag1\label{e-diss-def}\\
E_\mathrm{int} &amp;= 
  E_\mathrm{opt}[\ce{(XY3)2}] - 2E_\mathrm{frag}[\ce{(XY3)^{\neq}}] 
  %\ddag not implemented
  \tag2\label{e-int-def}\\
E_\mathrm{def} &amp;= 
  E_\mathrm{frag}[\ce{(XY3)^{\neq}}] - E_\mathrm{opt}[\ce{XY3}]
  \tag3\label{e-def-def}\\
E_\mathrm{diss} &amp;= 
  E_\mathrm{int} + 2E_\mathrm{def}\tag{1'}
\end{align}$$

Results &amp; Discussion

The Monomers $\ce{XCl3; X{=}\{B,Al\}}$.

Let's just get the obvious out of the way: Boron is (vdW-radius 205 pm) smaller than aluminium (vdW-radius 240 pm). For comparison chlorine has a vdW-radius of 205 pm, too. That is pretty much reflected in the bond lengths and the chlorine-chlorine distance.
\begin{array}{llrrr}\hline
&amp;\ce{X{=}}&amp; \ce{Al} &amp;\ce{B} &amp;\ce{Cl}\\\hline
\mathbf{d}(\ce{X-Cl})&amp;/\pu{pm} &amp; 206.0 &amp;173.6&amp;--\\
\mathbf{d}(\ce{Cl\bond{~}Cl'})&amp;/\pu{pm} &amp; 356.8 &amp; 300.6 &amp; --\\\hline
\mathbf{r}_\mathrm{vdW}&amp;/\pu{pm} &amp; 240   &amp; 205   &amp; 205\\
\mathbf{r}_\mathrm{sing}&amp;/\pu{pm} &amp; 126   &amp; 85   &amp; 99\\
\mathbf{r}_\mathrm{doub}&amp;/\pu{pm} &amp; 113   &amp; 78   &amp; 95\\\hline
\end{array}

From this data we can draw certain conclusions without further looking. The boron monomer is much more compact than the aluminium monomer. When we compare the bond lengths to the covalent radii (Pyykkö and Atsumi) we find that the boron chloride bond is about the length that we would expect from a double bond ($\mathbf{r}_\mathrm{doub}(\ce{B}) + \mathbf{r}_\mathrm{doub}(\ce{Cl}) = 173~\pu{pm}$). While the aluminium chloride bond is still significantly shorter than a single bond ($\mathbf{r}_\mathrm{sing}(\ce{Al}) + \mathbf{r}_\mathrm{sing}(\ce{Cl}) = 225~\pu{pm}$), it is still also much longer than a double bond ($\mathbf{r}_\mathrm{doub}(\ce{Al}) + \mathbf{r}_\mathrm{doub}(\ce{Cl}) = 191~\pu{pm}$).

This itself offers compelling evidence, that there is more &pi;-backbonding in $\ce{BCl3}$ than in $\ce{AlCl3}$. Molecular orbital theory offers more evidence for this. In both compounds is a doubly occupied &pi; orbital. The following pictures are for a contour value of 0.05; aluminium (left/top) and boron (right/bottom)




In numbers, the main contributions are as follows (this is just a representation, not the actual formula):
$$\begin{align}
\pi(\ce{BCl3}) &amp;= 
   21\%~\ce{p_{$z$}-B} + \sum_{i=1}^3 26\%~\ce{p_{$z$}-Cl^{$(i)$}}\\
\pi(\ce{AlCl3}) &amp;= 
   13\%~\ce{p_{$z$}-Al} + \sum_{i=1}^3 29\%~\ce{p_{$z$}-Cl^{$(i)$}}
\end{align}$$

There is still some more evidence. The natural atomic charges (NPA of NBO6) fairly well agree with that assesment; aluminium is far more positive than boron.
$$\begin{array}{lrr}
                    &amp; \ce{AlCl3} &amp; \ce{BCl3}\\\hline
\mathbf{q}(\ce{X})~\text{[NPA]}  &amp; +1.4       &amp; +0.3     \\
\mathbf{q}(\ce{Cl})~\text{[NPA]} &amp; -0.5       &amp; -0.1     \\\hline
%\mathbf{q}(\ce{X})~\text{[QTAIM]}  &amp; +2.4       &amp; +2.0    \\
%\mathbf{q}(\ce{Cl})~\text{[QTAIM]} &amp; -0.8       &amp; -0.7     \\\hline
\end{array}$$

The analysis in terms of QTAIM also shows that the bonds in $\ce{AlCl3}$ they are predominantly ionic (left/top) while in $\ce{BCl3}$ are predominantly covalent (right/bottom).




One final thought on the bonding can be supplied with a natural resonance theory analysis (NBO6). I have chosen the following starting configurations and let the program calculate their contribution. 



The overall structures in terms of resonance are the same for both cases, that is if you force resonance treatment of the aluminium monomer. Structure A does not contribute, while the others contribute to about 31%. However, when not forced into resonance, structure A is the best approximation of the bonding situation for $\ce{AlCl3}$. In the case of $\ce{BCl3}$ the algorithm finds a hyperbond between the chlorine atoms, a strongly delocalised bond between multiple centres. In this case these are 3-centre-4-electron bonds between the chlorine atoms, resulting from the higher lying degenerated &pi; orbitals.



This all is quite good evidence that the monomer of boron chloride should be more stable towards dimerisation than the monomer of aluminium.

The Dimers $\ce{(XCl3)2; X{=}\{B,Al\}}$.

The obvious change is that the co-ordination of the central elements goes from trigonal planar to distorted tertrahedral. A look at the geometries will give us something to talk about.

\begin{array}{llrrr}\hline
&amp;\ce{X{=}}&amp; \ce{Al} &amp;\ce{B} &amp;\ce{Cl}\\\hline
\mathbf{d}(\ce{X-Cl})&amp;/\pu{pm} &amp; 206.7 &amp;175.9&amp;--\\
\mathbf{d}(\ce{X-{\mu}Cl})&amp;/\pu{pm} &amp; 226.1 &amp;198.7&amp;--\\
\mathbf{d}(\ce{Cl\bond{~}{\mu}Cl})&amp;/\pu{pm} &amp; 354.1 &amp; 308.0 &amp; --\\
\mathbf{d}(\ce{{\mu}Cl\bond{~}{\mu}Cl'})&amp;/\pu{pm} &amp; 323.6 &amp; 287.3 &amp; --\\
\mathbf{d}(\ce{B\bond{~}B'})&amp;/\pu{pm} &amp; 315.7 &amp; 274.7 &amp; --\\\hline
\mathbf{r}_\mathrm{vdW}&amp;/\pu{pm} &amp; 240   &amp; 205   &amp; 205\\
\mathbf{r}_\mathrm{sing}&amp;/\pu{pm} &amp; 126   &amp; 85   &amp; 99\\
\mathbf{r}_\mathrm{doub}&amp;/\pu{pm} &amp; 113   &amp; 78   &amp; 95\\\hline
\end{array}

In principle nothing much changes other than the expected elongation of the bonds that are now bridging. In case of aluminium the stretch is just below 10% and for boron it is slightly above 14%, having a bit more impact. In the boron dimer also the terminal bonds are slightly (> +1%) affected, while for aluminium there is almost no change.

The charges are not really a reliable tool, especially when they are that close to zero as they are for boron. In both cases one can see that charge density is transferred from the bridging chlorine to the central $\ce{X}$.

$$\begin{array}{lrr}
                    &amp; \ce{(AlCl3)2} &amp; \ce{(BCl3)2}\\\hline
\mathbf{q}(\ce{X})~\text{[NPA]}  &amp; +1.3       &amp; +0.2     \\
\mathbf{q}(\ce{Cl})~\text{[NPA]} &amp; -0.5       &amp; -0.1     \\\hline
\mathbf{q}(\ce{{\mu}Cl})~\text{[NPA]} &amp; -0.4       &amp; +0.1     \\\hline
\end{array}$$

A look at the central four-membered-ring of in terms of QTAIM offers that the overall bonding does not change. In aluminium they get a little more ionic, while in boron they stay largely covalent.




The NBO analysis offers a maybe quite surprising result. There are no hyperbonds in any of the dimers. While a description in these terms is certainly possible, after all it is just an interpretation tool, it is completely unnecessary. So after all we have two kinds of bonds in the dimers four terminal $\ce{X-Cl}$ and four bridging $\ce{X-{\mu}Cl}$ bonds. Therefore the most accurate description is with formal charges (also the simplest). The notation with the arrows is not wrong, but it does not represent the fact that the bonds are equal for symmetry reasons alone.



To make this straight: There are no hyperbonds in $\ce{(XCl3)2; X{=}\{B,Al\}}$; this includes three-centre-two-electron bonds, and three-centre-four-electron bonds. And deeper insight to those will be offered on another day.

The differentiation between a dative bond and some other for of bond does not make sense, as the bonds are equal and only introduced by a deficiency of the used description model.

A natural resonance theory for $\ce{(BCl3)2}$ gives us a overall contribution of the main (all single bonds) structure of 46%; while all other structure do contribute, there are too many and their contribution is too little (> 5%). I did not run this analysis for the aluminium case as I did not expect any more insight and I did not want to waste calculation time.

Dimerisation - yes or no

The energies offer us a clear trend. Aluminium likes to dimerise, boron not. However, there are still some things to discuss. I am going to argue for the reaction 
$$\begin{align}
\ce{2XCl3 &amp;-&gt; (XCl3)2}&amp;
\Delta E-\mathrm{diss}/E_\mathrm{o}/H/G&amp;.
\end{align};$$
therefore if reaction energies are negative the dimerisation is favoured.

The following table includes all calculated energies, including the energy decomposition analysis mentioned at the beginning. All energies are given in $\pu{kJ mol^-1}$.
\begin{array}{lrcrcrcrr}
 \Delta &amp; E_\mathrm{diss} &amp;(&amp; E_\mathrm{int} &amp;+2\times&amp; E_\mathrm{def}&amp;)&amp; E_\mathrm{o} &amp;H &amp;G\\\hline
\ce{Al} &amp; -113.5 &amp;(&amp; -224.2 &amp;+2\times&amp; 55.4&amp;)&amp; -114.7 &amp; -60.4 &amp; -230.4\\ 
\ce{B}  &amp;   76.4 &amp;(&amp; -111.2 &amp;+2\times&amp; 93.8&amp;)&amp;   82.6 &amp; -47.1 &amp;  152.5\\\hline
\end{array}

The result is fairly obvious at first. The association for aluminium is strongly exergonic, while for boron it is strongly endergonic. While both reactions should be exothermic, stronger for aluminium, the trend for the observed electronic energies ($E_\mathrm{o}$ including the zero-point energy correction) and the (electronic) dissociation energies reflect the overall trend for the Gibbs enthalpies. 

While it is fairly surprising how strongly entropy favours association of $\ce{AlCl3}$, it is also surprising how it strongly disfavours it for $\ce{BCl3}$.
A look at the decomposed electronic energy offers great insight into the reasons why one dimer is stable and the other not (at room temperature).
The interaction energy of the fragments is double for aluminium than it is for boron. This can be traced back to the very large difference in the atomic partial charges. One could expect that the electrostatic energy is a lot more attractive for aluminium than it is for boron.
The deformation energy on the other hand clearly reflects the changes in the geometry discussed above. For aluminium there is a smaller penalty resulting from the elongation of the $\ce{Al-Cl}$ bond and pyramidalisation. For boron on the other hand this has a 1.5 times larger effect. The distortion also weakens the &pi;-backbonding, which the additional bonding would need to compensate.
The four-membered-ring is certainly not an ideal geometry and the bridging chlorine atoms come dangerously close.

Conclusion, Summary and TL;DR:

The distortion of the geometry of the monomer $\ce{BCl3}$ cannot be compensated by the additional bonding between the two fragments. Therefore the monomers are more stable than the dimer. Additionally entropy considerations at room temperature favour the monomer, too.

On the other hand, the distortion of the molecular geometry in $\ce{AlCl3}$ is less severe. The gain in interaction energy of the two fragments well overcompensates for the change. Entropy also favours the dimerisation.

While size of the central atom is certainly the distinguishing factor, its impact is only severe on  the electronic structure. Steric crowding would not be a problem when the interaction energy would compensate for that. This is quite evident because $\ce{BCl3}$ is still a very good Lewis acid and forms stable compounds with much larger moieties than itself.

References

The used van der Waals radii were taken from S. S. Batsanov Inorg. Mat. 2001, 37 (9), 871-885. And the covalent radii have been taken from P. Pyykkö and M. Atsumi Chem. Eur. J. 2009, 15, 12770-12779.

Computations have been carried out using Gaussian 09 rev D.01 with NBO 6.0. Additional analyses have been performed with MultiWFN 3.3.8. Orbital pictures were generated with the incredible ChemCraft.