You can think of difference equations as "discretized differential equations," or alternately you can think of differential equations as "difference equations in the limit as the difference goes to zero." This shows that they're very closely related in their definitions alone. The integral version of this correspondence is made precise by the Euler-Maclaurin formula.
Another sense in which they're closely related is the nature of how they act on polynomials. The derivative $\frac{d}{dx}$ acts on the basis $\{ x^n \}$ for the space of polynomials by $\frac{d}{dx} x^n = n x^{n-1}$. It turns out there is a similar basis $\{ n! {x \choose n} \}$ for the space of polynomials such that the forward difference $f(x+1) - f(x)$ acts the same way: the forward difference of $n! {x \choose n}$ turns out to be $n(n-1)! {x \choose n-1}$. Like the derivative, the forward difference also has a notion of Taylor expansion in terms of this basis. This is generalized by the theory of Sheffer sequences and related topics.
Finally, solving some differential equations is equivalent to solving some difference equations. For example, solving a linear homogeneous differential equation is basically the same thing as solving a linear homogeneous recurrence for the coefficients of the Taylor series. (This is because the derivative acts as left shift on Taylor coefficients, which also implies that the Wronskian and the Casoratian are really the same thing.)