The local vol model has exactly enough freedom to match the individual densities $X_t.$ There is no additional freedom in the local vol model to match even a joint density for a pair of times $(X_t,X_s).$
When you ask about the joint density across the continuum of times $t \in [0,T]$ it is pretty easy to show that any local vol model differs from any stochastic vol model provided at least one of the models has positivestochastic volatility model has nonzeeo vol-of-vol. That is because the joint density along the continuum of times gives full information about the measure on paths $X_t$.
For a local vol model, the instantaneous variance of a path $X_t$ at time $t$ is almost surely $\sigma^2(X_t,t).$ For a stochastic volatility model with nontrivial vol-of-vol this is not true for $t>0$: the instantaneous variance at time $t$ conditioned on $X_t$ is a random variable with mean $\sigma^2(X_t,t)$ but posive variance.