A B737-700 flight crew reported landing without clearance after a wake vortex encounter distracted them on final.

Narrative

I was the pilot not flying on this leg; which was the last leg (fourth) of the first day of our three-day. We were being vectored for the visual approach landing north in SEA. There were several other aircraft being vectored to parallel runways and we were following a B737 about five miles ahead. Approach did not assign a runway until the final vector to intercept final. They told us to follow traffic ahead and intercept 34R. We complied and about 30 seconds later they told us we were cleared for the visual 34C. I asked clarification on the runway assignment. They apologized and reconfirmed 34C. We were now three to four miles in trail of traffic ahead and the First Officer began to slow and configure. On about a six mile final and about three miles in trail; we flew into some moderate wake turbulence. The First Officer disconnected the autopilot and I asked him to fly slightly above the glidepath. We completed the Before Landing Checklist and the rest of the approach was uneventful. I did not catch the fact that we were not sent over to Tower. We still landed on Approach frequency and switched over upon clearing the runway. Tower cleared us across 34R and we taxied uneventfully to the gate. My attention was diverted with the runway changes and wake turbulence on final. I normally turn on the Outboard Landing lights with the clearance to land from Tower. As we cleared the runway and I took the aircraft; went to stow the speedbrake; and turn off the landing lights (outboards were not on); I realized we were not on Tower. The First Officer made the frequency change as we cleared.