Reading about the manifold calculus (or embedding calculus of Goodwillie-Weiss), we find that the derivative of a good functor $F:O(M)^{op}\to Top$ at $\varnothing$ is given by $$ F'(\varnothing):=\text{hofib}(F(B)\to F(\varnothing)) $$where $B$ is any open subset of $M$ diffeomorphic to a ball. Here, $M$ is a manifold and $O(M)$ is the poset of open subsets of $M$. The map $F(B)\to F(\varnothing)$ is induced by the inclusion $ \varnothing \subset B$.
To understand this definition and how it is analogous to the usual definition of the derivative of a function at $0$, authors often argue that taking the homotopy fiber somehow amounts to considering the 'difference' between spaces $F(B)$ and $ F(\varnothing)$, as if it would play the role of the space $ F(B)-F(\varnothing)$.
One reason this might be the case is by considering the sequence in homotopy groups $$ ... \to \pi_n(F'(\varnothing)) \to \pi_n(F(B)) \to \pi_n(F(\varnothing))\to ... $$which, according to the author, says that the space $F'(\varnothing))$ is obtained from $F(B)$ by removing cells of $F(\varnothing)$ so that $F'(\varnothing))$ indeed computes the difference of the two spaces 'in homotopy'.
I am struggling to make sense of this argument and I must say it doesn't make a lot of sense to me. I am trying to convince myself on simple examples how the homotopy fiber can compute a difference between spaces. Here are exxamples where I know what the homotopy fiber is :
$\bullet$ If $p:E\to B$ is a fibration of spaces, then hofib$(p)\simeq M_p$ is the mapping fiber of $p$.
$\bullet$ If $\iota : A \hookrightarrow X$ is an inclusion, then hofib$(\iota)$ is space of paths in $X$ relative to $A$. In particular, $$ \text{hofib}(\star \to X) = \Omega X $$In both these examples, the homotopy fiber doesn't really seem to compute a 'difference' of spaces. What am I missing to make sense of this interpretation?
The homotopy fiber measures how far a map is from being an equivalence. The closer the homotopy fiber $X \rightarrow Y$ is to being $X \times \Omega Y$, the closer the map is to being trivial on homotopy groups. If the homotopy fiber is trivial, then the map is a weak equivalence. In some respect it is like a metric. In your example of a fibration, it is saying that the difference between the total space and the base space is essentially the fiber. This makes sense because squashing the fiber via projection gives you the base space.