Can an integral equation always be rewritten as a differential equation?

In general, no. An integral equation can be non-local, whereas a differential equation is local (in the sense that it can be described by a function over the jet-bundle). As an illustration

Let $K(x) = \delta_0(x) + \delta_1(x)$ be an integral kernel, where $\delta_i$ are the Dirac delta's supported at $i$. Consider the integral equation, for some fixed smooth $f$ $$ f(x) = \int K(x-y) \phi(y) dy $$ for the unknown $\phi$. The equation reduces to $\phi(x) + \phi(x+1) = f(x)$. Any continuous function $g(x)$ on $[0,1]$ satisfying $g(0) + g(1) = f(0)$ generates a continuous solution of the equation. I challenge you to find a differential equation whose solution set can be thus generated.

While I second Deane's comment that the author should be a bit more specific about the kind of equations he is interested in, in general the answer is no for integral and, more broadly,integro-differential equations. However, the latter can be reduced to functional-differential equations rather than to purely differential ones. For more details, see Section 6.6 of the book Symmetries and Conservation Laws for Differential Equations of Mathematical Physics (unfortunately the relevant pages appear to be excluded from Google preview).

For instance, I greatly doubt that one could reduce the Smoluchowski coagulation equation from Example 6.5 of the above book to a differential (as opposed to functional-differential) equation or system thereof.

With Charles Matthews comments in perspective, these are some notes I made sometime ago on this topic. I dont have the books in front of me so I can't look up the details right now.

1) In Zabreyko's book Integral equations (902860393X), there is the method based on Green's functions in Appendix A.

2) Bellman in Perturbation techniques Sec 10 points out that the other way (ODE to integral equation) is actually better

Conversion of differential eqn to integral equation is one of the powerful devices in approximation theory. Its potency is due to the fact that integration is a smoothing op, while differentiation accentuates small variations. If u(t) and v(t) are close together, then ∫u(s)ds and ∫v(s)ds will be comparable in value, but du/dt and dv/dt may be arbitrarily far apart. Consequently, when carrying out successive approximations, we prefer integral operators to differential operators. On the other hand, in numerical solutions, we prefer differential operators to integral operators.

3) You can also look up Handbook of Integral Equations by Polyanin. Sec 8.4.5, Sec 9.7 and sec 9.3.3 are three situations where the method reduces a specific integral equation to an ODE