Minnesota's Elderly Waiver and long-term-care Medical Assistance for a nursing home both generally use the same aged/blind/disabled financial eligibility pathway -- a spend-down income test and asset limits commonly cited as $3,000 single / $6,000 married-both-applying (unverified against a live DHS source, confirm before relying on it) -- but they pay for entirely different settings: EW for care inside assisted living or at home, nursing home MA for a licensed nursing facility.
The entitlement difference that matters most
Nursing home Medical Assistance is an entitlement -- once a resident is financially and medically eligible, the state must pay for a nursing home bed. Minnesota's Elderly Waiver is not: availability can depend on county allocation and case management capacity, and current waitlist/allocation status was not confirmed during this site's research, so ask your county human services agency directly rather than assuming a waiver spot is guaranteed once you're financially eligible.