Some ideas that may help
Help them understand the point of any difference in approach
If the emphasis turns to proof or derivation from more fundamental principles, you can make the analogy of "you know how to drive a car, let's learn how the engine works". They can argue that being a better driver makes you a better mechanic and visa-versa but they can't argue that knowing one means you know the other. This analogy exemplifies the link and the difference. Calling it all "motoring" isn't helpful in the first instance.
Name topics or even whole courses to more clearly indicate new content
A shift in direction often causes tension in the students, where they felt on solid ground with applying memorised rules in predictable contexts, but are very uncomfortable questioning why or delving into how, especially if this seems needlessly complicated to them.
In this case it's important for them to understand they're studying something other than what they were before (the engine rather than the driving) Flagging this up with language and explaining what's happening reduces the upset that students feel when the thing they're good at morphs into a different beast and also the overconfidence that comes from the familiarity of some of the answers.
I'd advocate calling topics like differentiating by taking limits things like "differentiation under the hood", "the truth behind differentiation" or similar phrases emphasising slightly more esoteric content. The more you can make clear what more you're expecting than their existing knowledge, the better you have communicated the content.
Acknowledge prior learning where appropriate
Taking the same calculus with limits example, you can satisfy the students' desire to have their existing knowledge recognised by saying "but of course you know what the final simplified form of the derivative will be from the rules you memorised at high school, so you'll be able to quickly and easily check your final answer with high school math".
Where the content is recap, making this clear can also lend more credibility to the claims of newness elsewhere.
Examples from my experience lecturing in the UK
Taking some examples from UK undergraduate courses with which I'm more familiar, and where the topic is rather radically different to what they learned at high school, you could call highlight this by calling Algebra 1 "Abstract Algebra 1" or even just "Sets, Groups, Rings and Fields 1".
(I used to work at one of the UK's top mathematics departments where we ran a similarly titled course in the first term for undergraduates with the specific aim of helping them to make the transition to university mathematics. There was some resistance at first, including to the name, but it was an effective course.)
At the end of the course you can then admit that at university we collectively call Group Theory, Ring Theory, Field Theory etc "Algebra", and say that it's strange that we use the same name, but you can see there's a sort of link there somewhere.
"Calculus" could be (as appropriate) "Multidimensional Calculus" or "Applied Real Analysis", as specific as you can be, again flagging with some words that there's something very new about this.
(edited version of answer to correct out some assumptions I'd made based on UK rather than US undergraduate courses)