A car seat base stays permanently installed in the car (via ISOFIX or belt), so the seat itself just clicks in and out without you re-fitting a seatbelt every trip. If you’ll be moving the seat in and out of the car often, a base is worth the extra cost purely for convenience and reduced fitting errors.
Pushchair adapters let a compatible infant car seat click directly onto your pushchair chassis, turning it into a "travel system" mode for napping transfers. Adapters are specific to a car seat + pushchair combination, so always check compatibility both ways before assuming a seat will fit your existing pushchair, or vice versa.
If you’re buying the pushchair and the car seat from different brands, search the exact model names together (e.g. "Brand X pushchair Brand Y car seat adapter") before you buy either, since not every combination is supported.