Proceed is almost always accompanied by an adverb or a preposition - proceed to, proceed with, proceed when, proceed after, etc. The common way of saying it is "Let's proceed with our chat." Prepositions are used after proceed when it is used in this manner - here are some examples: https://lingohelp.me/preposition-after-verb/proceed-with-or-proceed-to/ "Let's continue our chat." is a less formal way of saying the same thing, but can only be used after you've already begun chatting.

"Continue" suggests that you have already started something. "Proceed" would be used when you are starting from the beginning. If you had chatted with this person before, then "continue" would be correct (even if this was a new conversation)

To proceed and to continue means technically the same thing. While "continue" is to be more used when doing simple talks like moving. "*After break* Let's continue moving the boxes in" and Proceed is a more formal way of saying "Let's continue with the task at hand" like this.. "Let's proceed with our investigation" I hope this makes the difference a little clearer. It's not that big of a difference. But I would recommend using Continue to everything until someone says "I don't understand" and then you can use "proceed" as a synonym.

Oh! I know! "proceed" is an intransitive verb. It cannot take a direct object. That is the difference. EDIT: You can get around this problem by following it with a preposition (with, on, after, in, etc). Also, proceed sounds quite formal.