Yes, there are multiple benefits to completing all research tasks.
By perfecting the research, you get a huge boost to encounter a shiny for that species. The shiny rate becomes a base 1/1024(Source)
chance for standard encounters rather than 1/4096, along with other factors to further increase it
As Joe stated in their answer, performing the tasks to perfect a Pokémon's research will continue to grant research points to achieve 10 Stars in the Galaxy Team Rank. Brade also mentioned in a comment that simply achieving rank 10 for each Pokémon was not enough to reach 10 stars, so continuing (albeit not necessarily perfecting) these tasks is required to reach max rank.
As Pinckerman stated in their answer, the Pokéball icon that appears next to the Pokémon in the Pokédex turns to a shiny Pokéball.