One important thing to note is that both the time step and space step have default values (both of these are 1/20 the spatial interval). These are suitable for obtaining a rough estimate of the solution, or a fair estimate if the problem is rather simple.

For the standing wave problem in question, you'll probably want to decrase both of these.

I will shortly attach a modified version of the original worksheet pdsolve_exercise_damping_ini_linear_a.mw as pdsolve_exercise_damping_ini_linear_a_mod2.mw where I have made the following changes v.s. the original:
1) Changed the value of T0 from 2*Pi to 4*Pi/3 to agree with the finite difference worksheet.
2) Modified spacestep to be 2*Pi/500 (also to agree with the finite difference worksheet), and timestep to 1/200
3) Changed the chosen plotting times to 0, 0.053*T0, 0.105*T0, 0.158*T0 and 0.211*T0, to agree with the finite difference worksheet
4) Included an animation out to T0, and commented out the 3d plot (to reduce the worksheet size)

With these modifications, pdsolve provides a solution in excellent agreement with the finite difference approach.

I believe that the behavior of the solution is fully correct, as the problem in question can be exactly solved through separation of variables, and I added this at the end of the worksheet.

pdsolve_exercise_damping_ini_linear_a_mod2.mw