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Running the base_local_planner with dwa=true isn't deprecated, its a valid configuration for the planner and should work. However, if you're considering using DWA, its probably worth checking out the dwa_local_planner package. The dwa_local_planner is a cleaner implementation of the algorithm (from a code structure standpoint) that includes other goodies like optionally scaling the robot's footprint as it drives and supporting dynamic_reconfigure.

As explained on the wiki, the difference between DWA and TR lies in the velocity space they explore while rolling out trajectories. DWA has a more restricted space since it only considers velocities that can be reached over one simulation step, corresponding to the controller_frequency parameter which is used to calculate the sim_period variable you've referred to. TR, on the other hand, explores velocities achievable over the entire trajectory simulation specified by the sim_time parameter. So, if controller_frequency is set to 10Hz and sim_time is set to 1.5 seconds, DWA will explore the velocities reachable (given the acceleration limits of the robot) in the next 0.1 seconds, while TR will explore velocities reachable in 1.5 seconds.

The original DWA paper, I believe, scores trajectories based on the robot's heading relative to a goal point, distance to obstacles, and velocity. The WG implementation scores based on distance to a goal point, distance from a planned path, and distance to obstacles. There are a number of papers on using DWA in conjunction with global plans to avoid local minima, but I'm not positive that any of them use this exact method for scoring. I do know that the TR paper linked from the base_local_planner wiki page uses this scoring method, and that's where it comes from.