Suppose for the sake of contradiction that $|\Delta(m, n)| \le N$ for all $m, n$. Note that $f$ is injective, as
$$f(m) = f(n) \implies \Delta(m, n) = 0 \implies m = n,$$
as desired.

Let $G$ be the “arrow graph” of $f$, which is the directed graph with vertex set $\mathbb{N}$ and edges $n \to f(n)$. The first step in the solution is to classify the structure of $G$. Injectivity implies that $G$ is a disjoint collection of chains (infinite and half-infinite) and cycles. We have the following sequence of claims that further refine the structure.

Claim — The graph $G$ has no cycles.
Proof. Suppose for the sake of contradiction that $f^k(n) = n$ for some $k \ge 2$ and $n \in \mathbb{N}$. As $m$ varies over $\mathbb{N}$, we have $|\Delta(m, n)| \le N$, so $f^{f(n)}(m)$ can only take on some finite set of values. In particular, this means that
$$f^{f(n)}(m_1) = f^{f(n)}(m_2)$$
for some $m_1 \ne m_2$, which contradicts injectivity.

Claim — The graph $G$ has at most $2N + 1$ chains.
Proof. Suppose we have numbers $m_1, \dots, m_k$ in distinct chains. Select a positive integer $B > \max\{f(m_1), \dots, f(m_k)\}$. Now,
$$|\Delta(m_i, f^{B-f(m_i)}(1))| \le N \implies |f^B(1) - f^{f^{B-f(m_i)+1}}(m_i)| \le N.$$
Since the $m_i$s are in different chains, we have that $f^{f^{B-f(m_i)+1}}(m_i)$ are distinct for each $i$, which implies that $k \le 2N + 1$, as desired.

Claim — The graph $G$ consists of exactly one half-infinite chain.
Proof. Fix some $c \in \mathbb{N}$. Call an element of $\mathbb{N}$ bad if it is not of the form $f^k(c)$ for some $k \ge 0$. It suffices to show that there are only finitely many bad numbers.
Since there are only finitely many chains, $f^{f(c)}(n)$ achieves all sufficiently large positive integers, say all positive integers at least $M$. Fix $A$ and $B$ such that $B > A \ge M$.

If $f^{f(c)}(n) \in [A, B]$, then $f^{f(n)}(c) \in [A - N, B + N]$, and distinct $n$ generate distinct $f^{f(n)}(c)$ due to the structure of $G$. Therefore, we have at least $B - A + 1$ good numbers in $[A - N, B + N]$, so there are at most $2N$ bad numbers in $[A - N, B + N]$.
Varying $B$, this shows there are at most $2N$ bad numbers at least $A - N$.
Let $c$ be the starting point of the chain, so every integer is of the form $f^k(c)$, where $k \ge 0$. Define a function $g: \mathbb{Z}_{\ge 0} \to \mathbb{N}$ by
$$g(k) := f^k(c).$$
Due to the structure of $G$, $g$ is a bijection. Define
$$\delta(a, b) := \Delta(f^a(c), f^b(c)) = g(g(b+1)+a) - g(g(a+1)+b),$$
so the conditions are equivalent to $|\delta(a, b)| \le N$ for all $a, b \in \mathbb{Z}_{\ge 0}$ and $\delta(a, b) \ne 0$ for $a \ne b$, which is equivalent to $g(a+1) - a \ne g(b+1) - b$ for $a \ne b$. This tells us that $g(x) - x$ is injective for $x \ge 1$.

Lemma
For all $M$, there exists a nonnegative integer $x$ with $g(x) \le x - M$.
Proof. Assume for the sake of contradiction that $g(x) - x$ is bounded below. Fix some large positive $K$. Since $g(x) - x$ is injective, there exists $B$ such that $g(x) - x \ge K$ for all $x \ge B$. Then $\min\{g(B+1), g(B+2), \dots\} \ge B + K$, while $\{g(0), \dots, g(B)\}$ only achieve $B+1$ values. Thus, at least $K-1$ values are not achieved by $g$, which is a contradiction.
Now pick $B$ such that $g(B) + N \le B$ and $g(B) > N$. Note that infinitely many such $B$ exist, since we can take $M$ to be arbitrarily small in the above lemma. Let
$$t = \max\{g^{-1}(g(B) - N), g^{-1}(g(B) - N + 1), \dots, g^{-1}(g(B) + N)\}.$$
Note that $g(t) \le g(B) + N \le B$, so we have
$$|\delta(t-1, B - g(t))| = |g(B) - g(t-1 + g(B+1 - g(t)))| \le N,$$
so
$$t - 1 + g(B + 1 - g(t)) \in \{g^{-1}(g(B) - N), g^{-1}(g(B) - N + 1), \dots, g^{-1}(g(B) + N)\},$$
so by the maximality of $t$, we must have $g(B + 1 - g(t)) = 1$, so $B + 1 - g(t) = g^{-1}(1)$. We have $|g(t) - g(B)| \le N$, so
$$|(B - g(B)) + 1 - g^{-1}(1)| \le N.$$
This is true for infinitely many values of $B$, so infinitely many values of $B - g(B)$ (by injectivity of $g(x) - x$), which is a contradiction. This completes the proof.