All molecular simulation force fields are essentially "fitted" mathematical functions. The parameters are adjusted until the simulation results match some physical properties we care about. As Geoff says, Lennard-Jones force fields are not particularly realistic, in the sense that you wouldn't expect a simulation of atoms interacting solely through Lennard-Jones potentials to behave exactly like a real system. This becomes very important for things like proteins, where conformation and charge-charge interactions are very important.
However, what might not be clear from the other answers and comments is that all force fields are not 100% realistic. The trick is to intelligently choose your level of realism. It could be that a Lennard-Jones force field that ignores charge-charge interactions works perfectly well for your application. Without knowing the details, I can't tell you how suitable it would be.
I do know that for polymers all sorts of real phenomena can be modeled with very simple Lennard-Jones-like force fields. A famous example is this article by Kurt Kremer and Gary Grest:
Dynamics of entangled linear polymer melts: A molecular‐dynamics simulation. J. Chem. Phys. 92, 5057 (1990)
Also, as Geoff mentioned, LJ potentials work pretty well for things that can be modeled by the van der Waals equation of state: noble gases, simple diatomics, and other small molecules at high temperature and low pressure.
Some coarse-grained simulations use LJ potentials for each "blob," and depending again on what you want to study, this can be very successful.
And while we are on the subject of unrealistic force fields, the hard-sphere model has been used to successfully model things like glass transitions (I don't have a good representative link for this, but you can search for "hard sphere glass transition" to find many studies).
So, in response to your actual question:
Does it even make sense to differentiate between different materials when using a crude approximation like the Lennard-Jones potential?
The answer is:
Maybe - it depends on what the goal of your simulation is. What physical behavior are you trying to model? If you just want to see how a string of soft, slightly sticky beads sticks to another string, then you don't need to differentiate between atoms. If you want to see what happens when some of those beads are really sticky (sort of like hydrogen bonding sites on a protein), then it would make sense to use different parameters for very sticky vs. slightly sticky beads. It probably would never make sense to worry about which beads were carbon and which were oxygen, nitrogen, etc., or to try to select parameters for them that were based on published values (unless those studies were specifically looking at the same system and properties that you are). Most LJ parameters are derived from fitting simulated phase behavior to real phase behavior - which is pretty far removed from what you are doing.
If you want to get a better idea of what real proteins would do, then the problem becomes much more complicated. Real protein behavior has a lot to do with interactions with water, which means you would need to simulate that as well. Charge-charge interactions also become much more important (and the simulations much more computationally expensive)
My advice (and I think what the other responses are saying as well) is to choose your system and goals carefully so that you don't need this level of realism. Or, if you do need it, then do it right; don't try to make a simple model work for a system that it doesn't fit.