Can I offer a slightly contrary opinion to most of those offered here?
For years I only practiced on the kit. As an untrained drummer with frequent access to a full kit and sound-dampening rehearsal space, I just sat on the kit and muscled my way through learning, because I had the time/freedom to do so.
What I found out later was that by only using the full kit, I was actually hiding my technique behind big drum sounds. I was doing a lot of instinctive drags on the snare, and there was a lot of sloppy rebounding on the bass drum - but in the context of a full kit playing with a full band, the sound of those errors was minimized, and therefore forgivable.
It wasn't until I started recording in nice studios and attempting to play electronic kits that I realized how horrible my technique really was; close-miking and e-drum pads are almost humiliatingly unforgiving, and my sloppiness was amplified. I was horrified.
Around the same time, purchasing a practice pad was almost happenstance for me - I only got one because I felt I wasn't practicing enough in between tours, and living situations prohibited full drum volume. But I immediately realized, playing at that lower volume and hearing each individual strike without any ambient noise to mask them, that my poor technique was just as evident on the pad as it had been in the studio and on the e-kits (especially when practicing on the hard rubber side of the pad, where drags and ghost notes have to be extremely precise to sound right).
Nowadays I still practice on the pad during the week (quiet neighborhood) then hit the full kit for band practice on the weekends. The pad work has kept my hands in great shape - so much so that I fashioned a similar setup to use with my bass drum pedal, and that's helped my foot technique immensely. I find that while there is not - and can never be - a perfect translation between what I do on the pad versus what I do on the kit, there are certain fundamentals that can be exercised on a drum pad and absolutely have a positive effect on the kit playing (internalizing tempos, keeping even fills, noodling out complex parts, etc.).
TLDR: Plenty of work on the drum pad has undeniably improved my playing on a full kit.