Although the two formations look similar, the pike square was developed in a very different tactical environment than the phalanx.
The phalanx and the maniple were developed in an environment where the primary weapons were swords, spears, and occasionally slings. Cavalry was rare, and was typically light cavalry used as skirmishers or to protect an army's flanks. Infantry was the supreme fighting force, and the maniple and phalanx were both developed as a counter to infantry.
The phalanx and the maniple were both employed offensively: to push against the enemy's formation and break it.
The Renaissance, on the other hand, was the tail end of a period when heavy cavalry was the dominant force. Ranged weapons such as bows, cannon, and early handheld firearms were common, and infantry were considered a "yes, we've also got some of them" force by many commanders.
Tactically, the job of a pike square is to provide a defensive formation: to keep enemy cavalry away from your artillery and to keep enemy cavalry and infantry away from your arquebusiers (see: pike and shot). The job of breaking the enemy formation was given to the other elements of the army.