At the bottom right of the playing screen, there is a small banner that displays 3 Pokémon. Selecting it brings up a screen that details all the nearby Pokémon.
Previously caught Pokémon show in full colour, uncaught Pokemon show as a silhouette, and doubles can be displayed. When I select any of these Pokémon and click the footprints the Pokémon becomes circled in purple on the previously mentioned banner.
Why would I want to "watch" a Pokémon in this way?
I have observations and assumptions, so far, but nothing concrete.