NAT vs PAT?

I've been reading into NAT and PAT, and even saw a few videos on it but I still don't quite get it and it's confusing me a little bit. So from what I understand...

NAT translates the private IP to a public IP to access the internet, however it requires just as many public IPs for every private IP? I'm sure I'm not understanding it correctly, but if the above is true, what exactly is the point of NAT if you need a public IP for each private IP, how does that save IP addresses?

PAT makes more sense to me, from what I gathered... PAT works similarly except it is capable of hosting multiple private IPs using one public IP but instead uses port numbers such as 192.168.1.1:1000 and 192.168.1.1:1001 for example. So if that's true it seems like PAT should always be used as it seems to be far more efficient. I'm not sure what I'm missing here, would love some clarification on this, thank you in advance for taking the time to read and respond.