In the team I am on, our idea on the matter is as follows:
Continue is used when you're talking about a directed flow forward only. Continue implies that anything you've done hitherto will be saved, so that you can move forward in the workflow. Ideally in a Continue-based setup, there will be alternate ways to return to previous app states, if your design supports doing that.
Next is is also used in a directional-flow situation, and also implies saving progress, but is usually paired with a Back command of some kind that does the same thing, but in the opposite direction.
There is nothing offensive about using Next in place of Continue without a Back button. It's perfectly permissible, in fact, in most cases. However, Continue does a very very slightly better job of driving home that the user needs to progress in the workflow we are requiring of them.
Another factor, is in the absence of a Back button, Continue is preferable actually because of its size; it's easier to see because it's larger. Next is only half as long, character-wise and that's not even counting if the font is monospaced or not (pro-tip: it's usually not.) Users can, and will, get stuck, or worse, frustrated, merely because they can't see a control they need easily enough. Of course, this can be mitigated by good styling, ensuring that the button is large enough, and is a 'constructive' color like green, which indicates that information will be saved/preserved, and is a primary action (one of the few buttons on the page).
The copy mostly comes down to preference in this case.