There are many similarities indeed. Both uses a foliation (even though LQG people don't call it like that, they effectively define it) and both uses Regge Calculus as its math.
But huge difference, that LQG seems to be focusing on the local aspects of the interactions of the discrete building blocks of spacetime. As the theory is working with connections and gauges, it is taking a traditional field theoretical approach, and aims to combine gr with the standard model.
In the other hand the CDT action contains only global information, like the number of vertices and simplices in of given triangulation, multiplied with the Newton and cosmological constants (and a third one quantifieing the difference between space and time. The action is global and no dynamics is put in by hand. The result of the numerical simulations, the shape of the configurations are given by the interplay of the normalizing factor in the partition function and the entropy.
Phase diagram-wise they share similarities also. LQG has a fully connected state, which probably corresponds to the crumpled phase in CDT. (which is a phase where the time extent of the universe disappears and there is only space).
Huge difference between the to, that LQG can operate with strong couplings, meaning that there are concrete couplings between the loops and fields. In CDT there is week coupling only, meaning that everything happens via the geometry.
LQG describes a chunk of spacetime, you can perform LQG calculations on a few simplices. Actually there are attempts to put calculations to qbits. (For example here).
CDT is a background independent approach, gluing the simplices creates the spacetime itself. Due to finite size effects/ Lattice artefacts it dies not make sense to describe CDT with a few simplices. This is why during numerical simulations a few 100k (or even million) simplices are used.
A CDT universe in the C phase (de-Sitter) describes a quantum universe with good semiclassical behavior and proper volume fluctuations on the top of it in accordance with the Hartle - Hawking minisuperspace model, where the only dynamical parameter is the scale factor. Furthermore, most recent finding shows, that the initial inhomogeneities of the universe and Dark Matter has probably quantum gravitational and not particle like origin (see Cosmic Fibers from quantum gravity ).
Even though they describe a 4 dimensional spacetime, both theories include (rather results) a dimensional reduction. At close distances spacetime behaves as it was effectively 2 dimensional.
LQG is a huge field, planck sized calculations are done, also tensor networks can be allied to it, and there is a loop quantum cosmology attempting to describe the early universe. I think that Dark Matter for example would be of particlish origin there.
Also, if I'm correct the LQG description contains gravitons, while CDT is a purely geometric approach.
In LQG you can take a chunc of spacetime, and calculate the dynamics of it, evolve it in time.
In CDT the whole spacetime history is calculated with fixed time extent. Every configuration of CDT is a closed history of the evution of the 3 dimensional submanifold from start to end (represented by the foliation, start and end is typically the same because the topology of the euclidean time direction is a circle, the snake bites its tail for numerical reasons ).
