Q:

Why can I sign but not encrypt?

-edit- After following a chain of links from this question I see this is similar/duplicate of Why should one not use the same asymmetric key for encryption as they do for signing?  but this question is asking if there is a technical reason why we shouldn't and the linked is less technical and not exclusively to rsa
I may have the wrong understanding. GPG lets you generate a RSA key that is sign only. From my understanding an RSA key is a public and key pair. Signature works by hashing the message and encrypting it with the private key and others decrypts it using the public key. From my understanding if the hash matches what is decrypted than the message is signed.
How can GPG generate a key and sign it but not let me encrypt messages? I believe in another app it suggested you have one key for signing and one key for encryption but there was no reason why. Wouldn't that be confusing because an encrypted message and a signed cleartext would be using different keys and appear to be from a different user?

A:

Please don't use the "explanation" of signatures as "encryption with the private key" because it is a flawed analogy that does not actually work.
RSA is really two algorithms, one for asymmetric encryption, and one for signatures. It so happens that the two algorithms have some common mathematical elements, and, in particular, use the same kind of key (which is why we call it a "RSA key", not a "RSA encryption key" or "RSA signature key"). Thus, it is conceivable that a given RSA key pair may be used for both encryption and signatures.
However, there are good reasons why you would like to keep encryption and signature keys separate. The OpenPGP format acknowledges that, and thus "tags" keys with an indication of their intended usage. Therefore, your signature-only RSA key is a RSA key that OpenPGP tagged with a label that mostly says "please don't use that key for encryption".

A:

Why do apps encourage you to use separate key-pairs for encryption and for signing?
You are correct that there is no technical reason why you can't use one RSA key-pair for both encryption/decryption and signing/verification. But just because the technology will allow it, does not mean it's a good idea. The reason why has to do with backups and what happens if you lose your private key (or have it stolen). I think the best way to explain this is to walk through some example cases.
For the following, let's say that a 1-keypair user is someone who uses the same keypair for encryption and signing operations, while a 2-keypair user is someone who has two separate keypairs attached to their profile.
For these examples, let's also say that you have a bunch of important files on your hard drive that you've encrypted for yourself, and also a bunch of documents that you've signed and sent to other people.
Case #1: 1-keypair user, you lose / delete your private key
You can no longer decrypt the files on your hard drive. All those things that were important enough to encrypt? They're gone, you'll never get them back. This may sound silly, but people lose their keys all the time when they lose their phone / laptop dies / they forget the master password, etc. With GPG there's no way to recover it.
As for signatures, no problem, the documents you've already signed will still verify properly because the paired public key is still fine, and for new documents you can just make a new signing key.
You may think "Alright, then I'll backup my private key on dropbox or something", well consider Case #2
Case #2: 1-keypair user, your private key gets stolen by a hacker
When you put your data on cloud storage, you are allowing the admins of that server to access it. So some other person (who's not you) now has access to your private key - at least in theory. So what? Well, if they ever get your laptop, they can decrypt those important files. Also, and more importantly, they can now sign documents in your name.
This may not be a big deal for you, but since digital signatures are legally binding in some places, it's a huge deal for politicians and lawyers who don't want hackers singing legal contracts on their behalf. In law, there is a concept of non-repudiation which means that if a document bears your signature then we know that you were the person who signed it. As soon as more than one person has access to your private key (even in theory), you can no longer claim non-repudiation, which means, in the worst case, that any contract you've ever signed with that key becomes void.
These two cases show that:

You want to back-up your decryption private key so that there's no risk of losing your data. In fact many companies and government organizations require their employees to back-up decryption keys on a central server.

You never ever Ever EVER EVER want to back-up a signing key. Ever.

The Solution
For 1-keypair users you have to choose between protecting your data, and protecting your non-repudiation. Meanwhile,2-keypair users don't have that problem, and since there's really no downside to having a second keypair in your profile, there's no reason that anyone should ever be a 1-keypair user.

