TL;DR. Two premium airline cards with lounge access at the core. AAdvantage Executive ($595) includes Admirals Club membership (full access for cardholder and authorized users), earns 4x on American, and adds 2024-refresh credits of roughly $360/year in Lyft, Grubhub, and Avis credits. Delta Reserve ($650) includes Sky Club access for the cardholder and the upgraded First Class/Comfort+ companion certificate. Pick based on which airline you fly: American (Executive) or Delta (Reserve).
The three dimensions that actually decide it. First, lounge network. Admirals Club is more numerous in the U.S. (50+ locations); Sky Club is similar (~50). Both extend internationally via airline partner lounges. Second, companion certificate. Reserve has one (upgraded class); Executive does not include a companion cert. Third, fee. $55 difference. Executive is slightly cheaper.
Real customer scenario for each. If you fly American Airlines and use Admirals Club access for yourself plus authorized users (the AUs get access too), Executive's $595 covers everyone in the family at a single fee. If instead you fly Delta with a partner annually, Reserve's companion cert is a single-trip value of $500-1,500.
The trap to avoid. Holding both. The lounge access does not stack across airlines (American flights vs Delta flights). Pick one based on which carrier dominates your travel.