Most consumer programs need both wallets. The customer experience is similar; the implementation differs.
Customer experience
Both wallets store passes on the phone, support barcodes and QR, and can show lock-screen notifications when passes update or when relevant locations apply.
Platform coverage
Apple Wallet is standard on iPhone. Google Wallet is standard on Android. Dual publishing avoids excluding half your audience.
Technical differences
Apple uses signed .pkpass bundles and PassKit web services. Google uses Wallet API classes and objects. Unified platforms hide that split.
Program design implications
Design templates that read well on both form factors. Test fonts, logo safe zones, and field lengths on real devices.
Recommendation
Ship both wallets from one template and measure install split by traffic source.