While there is a logical symmetry between differential equations and integral equations, it seems that (as they say) "laws of nature" are written in differential equations, not integral equations. As far as I can tell, rigorous study of integral equations arose in the late 1800's in part because of the relative difficulty of rigorous study of (especially partial) differential equations. That is, bounded operators (e.g., inverses of differential operators!), especially compact operators, were far easier to prove theorems about than unbounded operators (differential operators).
Nevertheless, people using differential equations throughout the 19th and 20th centuries often were not slowed-down by worries about mathematical niceties (and very often reached correct conclusions with or without "proofs" whose legitimacy could be understood in the context of the times).
That is, integral equations seem mostly to be of interest either as a (temporary?) substitute for differential equations, or as a technical warm-up for rigorous treatment of differential equations.
The relatively more-recent Schwartz' Kernel Theorem, Grothendieck's "nuclear spaces" (and pseudo-differential operators, singular integral operators, etc.) can certainly be viewed as extrapolations of ideas about integral equations, but it is my impression that people applying such ideas are also less finicky about distinguishing "generalized functions" (as in Schwartz' distributions) from classical functions, so may consider Schwartz' Kernel theorem as "obvious".
So, perhaps, we don't teach introductory integral equations along with introductory differential equations because the differential equations have the more direct applications, and because the greater ease of rigorizing integral equations is irrelevant.