I am looking at two (common) definitions:
Definition 1: An affine space is a triple $(A, V, +)$ with $A \neq \emptyset$, with $(V, +_V)$ a vector space, and with $+ \colon A \times V \to A$ such that the following conditions hold: $$\begin{align} \forall a \in A \ \forall v,w \in V \colon \quad & (a + v) + w = a + (v +_V w) \\ \forall a,b \in A \ \exists v \in V \colon \quad & b = a + v \\ \forall a \in A \ \forall v,w \in V \colon \quad & a + v = a + w \Rightarrow v = w \end{align}$$ For any $a,b \in A$, the $v$ in the second condition is unique (by the third condition) and is denoted here by $b - a$. The map $- \colon A \times A \to V$ thus induced is called subtraction.
Definition 2: A difference space is a structure $(A, V, -)$ with $A \neq \emptyset$, with $(V, +_V)$ a vector space, and with $- \colon A \times A \to V$ such that the following conditions hold that are known as Weyl's axioms: $$\begin{align} \forall a,b,c \in A \colon \quad & (c-b) +_V (b-a) = c - a \\ \forall a \in A \ \forall v \in V \ \exists b \in A \colon \quad & b - a = v \\ \forall a \in A \ \forall b,b' \in A \colon \quad & b - a = b' - a \Rightarrow b = b' \end{align}$$ For any $a \in A$ and any $v \in V$, the $b$ from the second condition is unique (by the third condition) and is denoted here by $a + v$. The map $+ \colon A \times V \to A$ thus induced is called translation.
On the one hand, it is well known that $(A, V, +)$ is an affine space with subtraction $-$ if and only if $(A, V, -)$ is a difference space with translation $+$. On the other hand, the relationship between homomorphisms of affine spaces and homomorphisms of difference spaces seems to be more complex.
Question: Am I making a mistake and is the relationship between these classes of homomorphisms in fact easy to describe, or is it in fact non-trivial (and has been studied)?
If the latter is true, this would be a strong reason, in my view, to not start with Weyl's axioms in a textbook definition of affine space.