CONNECT TO

Use connect to when you talk about a physical link between one object and another.

Make sure the printer is connected to your computer.
My Wi-Fi works but I’m not connected to the internet.

CONNECT WITH

Use connect with when you talk about a relationship, a non-physical link.

These are typical symptoms connected with appendicitis.
He immediately connected with his therapist.
She is extremely good at connecting with her audience.

This, of course, also applies to the noun connection.

A man is being heard by the police in connection with a burglary last night.

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