Shaykh Muhammad ibn Saalih al-Uthaymeen rahimahullaah in his ‘Majmoo` Fataawaa’ vol. 16 pge. 237 was asked the following question:
Q: What is the wisdom in taking a different path (when returning from the prayer) on the day of Eid?
A: The wisdom as it relates to us is:
(1) Firstly; adhering to the way of the Prophet sallAllaahu `alaihi wa sallam, as this (act) is from the Sunnah. [As occurs in the narration reported by al-Bukhaaree, no. 986]
(2) Also from the wisdoms is that in it is an apparent manifestation, and an announcement of the Eid prayer in all of the marketplaces of that land.
(3) Likewise from the wisdoms is that one can see the condition of the people of the marketplaces – from the poor people and other than them. [So that he can give charity to the poor people (that he meets) on the second path. And Allaah knows best. See page no. 223]
(4) They said that from the wisdoms is that the two paths will bear witness for him on the Day of Resurrection.