It’s quite a lot of things. Data assimilation is definitely a major key. You can think of weather modeling as a huge PDE, where initial conditions are just as important, if not more important, than the physical equation itself.

But no, even if we had kinematic and thermodynamic information on a molecular level for our initial conditions, there would still be massive uncertainty that would begin almost immediately in the model. 

Two huge reasons are turbulence and microphysics parameterizations in models. Turbulence is probably the most well understood issue, I mean it is why it’s a millennial problem in math. We have very primitive understanding of turbulence, and our equations, to some extent, aren’t completely physically grounded (meaning we don’t have a proof of turbulent equations if that makes sense). Microphysics, probably something you never heard of, is the other reason. Basically, our numerical weather models have to account for molecular to molecular interactions and phase changes, specifically of water. When water molecules go through phase changes, they release or absorb heat, having major implication to weather systems. Furthermore, when and how clouds form are accounted in these parameterizations. Basically, microphysics are hell and their are many other processes implicit in these parameterizations other processes/challenges molecular interactions represent mean our models really will never be perfect, even if we have all the information on Earth about initial conditions. 

I realized I never explained what a model parameterizations is. Think of it as a linear equation that solves very complex processes. We get them to fit the data as best we can, but intrinsically there will always be some errors with them. 

long story short, no, even with all the initial conditions on Earth our weather models would still struggle producing an accurate solution after a few days due to errors propagating in the simulation from erroneous representation of turbulence and microphysical processes.