I am looking for an intuitive explanation that explains the difference between absolutely and conditionally convergent (Riemann) improper integrals. Please note I understand how one goes about determining this difference and is not the question being asked here.
The case for series
To paraphrase from Wikipedia, absolute convergence is important in the study of infinite series since its definition is strong enough to have properties of finite sums that not all convergent series possess, yet is broad enough to occur commonly. One such property is the rearrangment of terms in an absolutely convergent series. Doing so does not change the value of its sum - something which is not true for conditionally convergent series. So in a way one can say absolutely convergent series behave "nicely" without undue "conditions" being placed upon them when summed.
The case for improper integrals
So what is the case for improper integrals? Does a rearrangement of terms in the case of infinite series perhaps correspond to a rearrangement of various intervals of integration if partitioned?