Everything worth knowing, nothing that isn't.
Reach and dependency
SMS works on every handset with no app or account; push requires your app installed, notifications permitted, and the device online with the OS push service reachable. For near-universal reach to anyone, SMS wins. See /answers/what-is-a2p-sms.
Cost and engagement
Push is essentially free per message but suffers low opt-in and high mute rates; SMS costs per segment but is opened and read at far higher rates. The right economics depend on how critical and time-sensitive the message is.
Best job for each
Use SMS for OTP, transaction alerts, and anything that must be seen; use push for timely nudges to already-engaged app users. See /answers/sms-for-payment-alerts and /answers/best-channel-for-otp.
Use them together
A resilient design tries push for app users and falls back to SMS when push is undelivered or the user has no app — so the message always lands. See /answers/sms-fallback-strategy and /products/omnichannel.