I think the term "affine bundle" is used for at least two things: (1) A map $p:Y\to X$ such that for some open cover (in your choice of topology) there are isomorphisms $p^{-1}(U)={\Bbb A}^n \times U$ --- just like you said. (2) A torsor for a vector bundle, i.e., like (1) but with the added condition that the transition functions are affine-linear.
In the situation of (2), it's a vector bundle exactly when there's a section (like any torsor). A simple non-vector-bundle-example is the complement of the diagonal in ${\Bbb P}^1 \times {\Bbb P}^1$, projecting onto one of the factors.
For (1), I don't know any general (non-trivial) criterion for such a thing to be a vector bundle. (Maybe because the group $Aut({\Bbb A}^n)$ is so complicated...) A simple non-example is the 2nd-order jet scheme $\mathrm{Hom}(\mathrm{Spec}(k[t]/(t^3)),{\Bbb P}^1) \to {\Bbb P}^1$. The fibers are ${\Bbb A}^2$, and there's a section, but it's not linear. I suppose one test is whether the sheaf of $O_X$-algebras $p_*O_Y$ admits a grading generated in degree one. (This fails for the jet schemes, though there is a natural grading by scaling $t$.)