Series[-q (c[x1]  - c[x2]), {x2, x1, 1}]
Exp[%] // Normal // Simplify
(Exp[%% // Normal]) // Simplify

SeriesData[x2, x1, {q*Derivative[1][c][x1]}, 1, 2, 1]
1 + q*(-x1 + x2)*Derivative[1][c][x1]
E^(q*(-x1 + x2)*Derivative[1][c][x1])

z = Series[-q (c[x1] - c[x2]), {x2, x1, 1}]
(* q c'[x1] (x2 - x1) + O[x2 - x1]^2 *)

Exp[z] // Normal // Simplify
(* 1 + q (-x1 + x2) c'[x1] *)

(Exp[z // Normal]) // Simplify
(* E^(q (-x1 + x2) c'[x1]) *)

Series[Exp[-q (c[x1] - c[x2])], {x2, x1, 1}]
(* 1 + q (-x1 + x2) c'[x1] *)