They're more or less synonymous, but at least in Spain, "obsequio" is used to a greater extent, when used for commercial purposes.
Nevertheless, we can see all its meanings in the dictionary with examples for better understanding:
☆ Regalo __________________
♦ What is given to someone without expecting anything in return, as a token of affection or gratitude (regalo de cumpleaños).
♦ Pleasure or satisfaction that is received (Esos postres son un regalo para el paladar).
♦ Comfort and rest a person seeks to himself (vive con gran regalo en su mansión).
♦ Something very cheap (Estos pantalones fueron un regalo, solo me costaron 10€).
☆ Obsequio __________________
♦ Gift (Tenemos un obsequio preparado para usted).
♦ Displays of affection or respect someone in order to please him (Pásese por nuestra oficina y recibirá su obsequio de bienvenida).
I really hope this helps you, Ilya. Cheers!