I have stayed at both the Waldorf and the "new" Hilton in Panama.
As of Oct 2014, the Executive Lounge at the Hilton has not opened. Thus, if you are Gold or higher, don't expect the benefits of the lounge. It should open "soon".
Breakfast was included for me at both properties, can't remember if it was a Gold benefit or if it was in the rate. Breakfast was slightly better at the Waldorf, but still great at the Hilton with lots of choices of eggs, bacon, fresh fruits and baked goods. I'm a big breakfast eater and enjoyed them both.
I have stayed at the Hilton twice, the Waldorf once, so I like the Hilton better. Hilton has opportunities for an ocean view, the Waldorf really doesn't, as it is set the block behind the water.
Waldorf had a more 'service oriented' feel to it, as the staff there remembered my name and welcomed me in to the hotel each time I came and went.
The Hilton has an annoying two elevator step process to get to your room. Take an elevator from the ground floor to the lobby, then a second bank of elevators to the room.
The Waldorf room had nicer furnishings and a more "tech" feel to it (lights, drapes, etc.) The Waldorf bathroom had a nicer shower than the Hilton.
Each stay at the Hilton I was given a corner room and told it was a Jr. Suite. It had a bathroom set up with a separate shower and soaking (not jetted) tub. Never used the tub.
Lobby at the Hilton has a bar with the opportunity to overlook the ocean, I don't remember the lobby bar at the Waldorf.
Some of the rooms at the Hilton have balconies (according to the website) but I have never seen one, and would probably be too hot to spend much time on one anyway.
Hilton is right on Avenida Balboa, and has better access to local taxis than the Waldorf. Both hotels have the more expensive and nicer hotel taxis immediately available whenever I wanted.
Hope this helps, and I'm happy to answer other more specific questions!