Both get somebody into your Circle. They differ in who does the sending, and in one thing that is easy to miss.
The email invitation
You enter their name and email, and we send them a private invitation. It is addressed to that person and nobody else can use it.
It holds their spot straight away. From the moment you send it, that person counts against the room in your Circle, even though they have not accepted yet. This is deliberate: it stops you inviting four people into two seats and having to disappoint two of them.
The private link
You create a link and send it yourself, by text, by your own email, however you normally reach that person. Useful when you know somebody does not read email, or you want to hand it over in person.
A link holds nothing until somebody actually uses it.
Which to use
Email when you have their address and want it done.
Link when you would rather send it in your own words, or you are not sure which address they use.
There is no difference once they are in. Same Circle, same access, same everything.