Expanding on what Bruno said, one reason to run multiple clients at the same time is that there are usually three science puzzles running at the same time. Players who want to be competitive in the Foldit rankings generally play all the puzzles.
After starting a puzzle with "hand folding", players typically use recipes to refine their solutions. Recipes like Tvdl enhanced DRW 3.0.2 and TvdL DRemixW 3.0.2 can run for a long time. On large puzzles like the current 1445 and 1446, they can easily run for days.
Recipes automate the tools – like shake, wiggle, rebuild, remix, mutate, bands, and cutpoints – that you use in hand-folding.
Different recipes may give different results. Most recipes have some randomness built in, so even running the same recipe on the same solution may produce different results. And recipes have like EDRW and DRemW have lots of settings, which again can produce different results.
While a recipe is running, you can minimize the client window. Most machines have multiple "logical processors", or "cores". On most puzzles, a minimized Foldit client running a recipe uses about 100% of one logical processor. Larger puzzles use noticeably more. So on "four core" laptop, I can usually run three minimized clients on the normal-sized puzzles (somewhere around 100 amino acids).
If you're running just a single client, it can't take advantage of the extra logical processors. They'll sit mostly idle.
If your Foldit client is not minimized, the display itself uses one logical processor. So running a tool or recipe with the window "up" can use two or more logical processors. (One gripe is that Foldit uses a logical processor when the window's up, even if nothing's happening with the protein.)
Another reason to use multiple clients is group play. When you join a group, you can play as an "evolver", working on solutions shared by other group members. The winning solutions are often from the evolver side of the competition.
Bottom line, with multiple puzzles and multiple recipes, not to mention soloist and evolver, there are lots of reasons to run multiple clients.