From the in-game help menu (Question Mark in top left corner, Management Section, Page 4 on vet research):
"Once a research chain has been completed, the vet can be left researching the animal, this is advanced research. This will provide a boost to a zoo's education rating."
Education ratings in turn help keep guests happier longer, which can lead to more donations and spending, and also help your zoo's reputation (which in turn helps bring more guests to the zoo).
I've also seen unconfirmed player theories of some other possible impacts (although again, I'm not sure if these are 100% accurate):
Some say that happier guests (which education and advanced research influences) are more likely to go deeper into your zoo, and can go further distances between destinations before they give up and turn around.
Some say that happier guests improves an animals star ratings, (and/or that seeing an animal with advanced research makes happier guests see more animals or travel to see more animals, and that the number of viewers can in turn help increase the star ratings of other animals or the zoo as a whole).
Some say that advanced research helps with breeding and fertility -- either directly, or as a cumulative side effect of one of the other benefits, such as stars, etc.
These cumulative impacts are a bit tougher to prove or disprove, since there are so many other factors involved. But it is worth remembering that education ratings aren't just one static number to achieve, but rather a game mechanic that is tied to several other interconnected game mechanics, at least marginally.
Presuming that I have enough money to hire enough vets, I like to do it. I like to do it for the animals nearest to my entrance first (so that guests get an immediate education/happiness boost as they come in), as well as any animals that I'm having trouble breeding (just in case), and high-priced animals (both to get a donation boost, and because they often have larger habitats that guests need to traverse to/around). Sometimes I even make enough money that I can advanced research every animal in the zoo.
One important note and potential downside: Vets who are doing research (including advanced research) tend to forget all of their other tasks. So make sure you have a good mix of "working" vets who actually care for animals, and dedicated research vets who you aren't relying on to keep the rest of the zoo functional.