Lightning is the technology, think: apple pay.

A lightning invoice is the qr code you generate for payment. You can usually specify the amount required and even a message, hence invoice. You can send it by email, show it on screen, or share it online. Once paid once, it's paid, done.

Zaps is a nostr feature, it's more like a type of payment: like donations are a type of payment that helps those in need, Zaps are a type of payment that publicises the payment on nostr for everyone to see and to help publicly identify which nostr posts were most popular.

Zaps tend to rely on a technology called lnaddress, where you get an email address looking identifier and people can send you funds over lightning. This allows you to pay someone without them asking for payment first, invoices are the opposite.

It's like a payment vs a tip.