I practice slow for a couple of reasons. The most common reason I do this is to start off with as little tension as possible. The second is a mixture of kinaesthetic and sonic, particularly in a piece with counterpoint--to both get it under my fingers and to work out the balancing of voices.
Last year I learned Chopin's Op. 28 no. 10. At first I tried to just will myself through it, but the sound was heavy, while the piece is marked leggiero. So I practiced it for a day almost absurdly slow with light finger staccato, pianissimo, letting my fingers relax after sounding each individual note. The next day I played nonlegato and focused on smoothness, first slow and then ramping up the tempo, working beat by beat. And the following day I put it together, and the result was so much better than when I started off playing close to tempo with a sound that wasn't acceptable. Building up from a place of relaxation helps me get the sound I want, and I'm not baking tension into my muscle memory for the piece.
For the second kind of slow practice, I'm current working on Émile-Robert Blanchet's Op 53 no. 1--an etude for the left hand solo that generally involves thirds played successively with fingers 3-2 and 4-1 while 5 accompanies the thirds with longer note values. There are a lot of different articulations--nonlegato, legato--some stretches of 9ths that have to immediately collapse in the allegro tempo to tighter positions, and a few jumps involved--all of which make smoothness and voicing difficult, so slow practice really helps me balance everything out and focus on precision. The actual writing of the piece involves pretty much the same figuration, so articulation really makes the different sections of the piece what they are, and it's easy to lose that when I get overconfident and just try to zoom through the whole thing. This kind of slow practice is less about the mechanics of playing and more about ensuring I can achieve the sound I want, so I have a good standard and awareness of nuances as I approach getting the piece to speed.