The act of actually recording the musician playing on the individual tracks has changed too, but some of the process is still the same. Unless you're recording the entire band live, you still have to record individual performances in a layer by layer process.
In our case, we laid down a metronome track to help keep us in time, and along with those tracks, we laid down some "dummy" guitar and vocals. From there we laid down the drum performance for each song, then the bass guitar and then the rhythm guitars.
After that we will record guitar solos, steel guitar, fiddle, vocals and any other instrumentation, peformance by performance and track by track.
It's all very time intensive, but a lot of fun at the same time.