One can learn to play the piano on an electronic or midi keyboard with closed headphones without waking up the roommate. Is there anything equivalent for drumming? Some sort of nearly silent drum pad or percussion set with midi out?
Yes, there are electronic drums.
There will be a tapping sound when playing. This will likely not disturb your neighbors, but your room mate might find it disturbing.
I believe that playing with brushes is problematic, but I'm not updated on the technical advancements of electronic drums.
Professional electronic drum kits are made by many musical instrument manufacturers including Roland, Yamaha, Alesis, ddrum, Simmons, and even the Zildjian cymbal company. The link in the previous sentence goes to a list of kits sold at Guitar Center.
These are all "silent" in that they make no sound acoustically (however you can hear the sound of your drum sticks striking the plastic drum trigger pads, which do not resonate). When you strike a drum trigger, the "brain" in the electronic kit plays digital samples of acoustic drums, or physically-modelled sounds that mimic the sound of acoustic drums. To hear these sounds, you need to plug in headphones or cables connecting to an amplifier and speakers.
While they can be used for silent practice, the main purpose of these drum kits is twofold: to enable a drummer access to a much larger range of drum sounds than can be found in an acoustic kit, and to enable concert performance without the need for using multiple microphones all over the kit to amplify the sound. Rather than needing microphones, you simply run cables from the "brain" directly into the PA system, while the drummer hears what he is playing through monitor speakers.
As per the previous answers, there are electronic kits that are effectively silent, insofar as you only get the sound of a stick striking a rubber pad or, in the cases of some e-kits, mesh heads.
However, I'm an acoustic drummer and I've found playing on electronic kits to be problematic: they're invariably fixed to a frame, so you can be limited in where you can position drums and cymbals; and the sizes are way too small - hitting an 8-inch pad is not the same as hitting a 14-inch snare drum.
What I did was purchase a cheap (£60) kit from eBay, then fitted the snare and toms with triple-ply mesh heads (link to eBay seller of mesh heads in classic sizes). I chose these because I'm a heavy hitter, so wanted something that would last more than a week. This means that I can strike the drums as hard as I like and they are (arguably) quieter than even the rubber pads on electronic kits.
I also stuffed the bass drum with a duvet and two pillows and covered the 'strike zone' with a dense rubber pad. This is louder than all the other drums, but is comparable in volume to hitting your hand on the desk.
This in itself is a 'silent' drum kit that I can happily play as hard as I like, and - crucially - feels exactly the same as my 'real' drum kit.
However, you don't have to stop there: I got some ddrum triggers, which clamp on to the rim of my drums and a Roland bass drum trigger from eBay, wired all of that into an Alesis Trigger I/O, which then connects via USB to my Mac. Then I can use any piece of software (I personally use BFD Eco) to spit out realistic drum sounds for me to play along to. This is usually referred to as an electro-acoustic kit, and is usually the reserve of hobbyists like myself.
It's worth noting that my particular set-up is perhaps a little convoluted and not without issues - if you definitely want some kind of foldback (rather than just playing an acoustically silent kit), you may want to look into a drum brain that will accept multiple jack/XLR inputs, although brains on their own can be expensive. All told, I cobbled together my setup for around £300.