Without getting into a bunch of the math, the hobbyist's answer is pretty much this:

Pretty much any calculation pr presumption made about an airplane's behaviour presumes, first, that airflow over, under, and around any surfaces is uninterrupted...that is to say, for example, that the wings are treated, for purposes of such calculations, as a solid unbroken surface.

A gap in any surface results in air not 'successfully" completing the trip, as it were, for a variety of reasons. Probably most significant are the disruption of the flow (in terms of both direction and speed) and the drag created both by the disruption and by the impact of the air with whatever "flat surface" might be exposed by the gap.

Why can/does this cause problems on some airplanes? Simple...it causes the surface in question to behave differently than designed. Perhaps it's a bit more draggy, thus raising the overall drag of the aircraft, which then raises the power required to overcome said drag, which of course has the resultant change in angle of attack for a given power setting, blah blah blah blah. In short...the plane 'ain't right" *heh*

How MUCH difference it'll make will depend on a great many things. How "slick" is the airframe supposed to be? What's the airplane supposed to do? how precise is the airplane in its 'correct" configuration? How skilled/experienced is the pilot?

Take a VERY precise sleek pattern airplane with an experienced pilot....even a small change to the airflow can make a notable difference to such a pilot.

On the other hand, as wzak suggested...a less experienced pilot on a more 'soft" airplane like a trainer? 1/8" gap probably won't EVER be noticeable...though covering the gap as suggested above is still good practice simply because it's good to be in the habit of 'getting it right", as it were.