If you want to increase the aerobic contribution to the workout, a jog recovery is best. By jogging instead of standing around, you are not letting your anaerobic system properly recover before the next interval. You will notice, on a workout like 400m repeats that if you jog say, 200m recovery between reps, your reps will be a bit slower, despite you putting forth the same effort. This is because your anaerobic system is not helping you as much as it would be if you were standing around inbetween intervals.
This said, there is also benefit to standing recovery when it comes to increasing muscular capability. If you are doing mile repeats or faster, it would serve you better to take some standing recovery so you are able to reproduce the appropriate pace on subsequent reps.