Air carrier flight crew reported Ground Conflict with another aircraft during taxi to gate.

Narrative

During taxiing to the gate in reduced visibility due to moderate snowfall and very slushy taxiway conditions; DEN airport was in 'full deice' taxi routes. After clearing Runway 34R; we were instructed by the Rower Controller to taxi via 'Foxtrot; left turn Zulu; halfway across Zulu; contact Ground 121.85.' After contacting Ground Control; was then instructed to taxi 'via Lima and Alpha November' to our gate. Approaching Taxiway Echo Delta on Taxiway Zulu; noticed an Airline Y aircraft that appeared to not be 'giving way' to us. I aggressively applied brakes as did the Airline Y airplane. The Airline Y airplane came to a stop short of our Taxiway Lima. Having not been instructed to 'give way'; we continued our taxi. We reported to Ground Control that we were 'not instructed to give way to Airline Y'; and their response was 'Roger'. The Airline Y aircraft also reported to Ground Control that they were never told to 'give way' to us.Loss of situational awareness by Ground Control; possibly complicated by the low visibility and rapidly changing airport conditions. Ground Control may have been relying on ground radar only due to visibility.Extra vigilance by all parties involved during taxi; especially during lower visibility taxi with contaminated surfaces. And when in doubt; clarify the ATC clearance; although we were under the impression that we had the right of way due to the taxi clearance we were given.