All three programs can simulate a drop test.  Transient Structural requires contact to be defined prior to running the analysis. Explicit Dynamics automatically detects body to body contact when the solution begins.

Explicit Dynamics and LS-DYNA both use an explicit solver, which means it takes very small time steps so that dynamic equilibrium is always maintained. Transient Structural uses an implicit solver, which means it can take large time steps but it must converge on the dynamic equilibrium at each time step and may fail to converge. 

Transient Structural and Explicit Dynamics are more tightly integrated to run inside Mechanical Workbench.  LS-DYNA is less well integrated into Mechanical Workbench and uses stand alone LS-Prepost and LS-Run programs.

Here is a list of YouTube videos on Explicit Dynamics: