Attacks on the Neutrals by the Axis generally don’t occur until you’ve secured a front which generally means Moscow has fallen and you’re driving for Cairo.
Taking Spain assists in this as if you eliminate the NB in Gib (or at least force a fight in Spain/Gib), the Allies cannot advance into the Med without controlling it. This neuters the Allied 1-2 punch to some extent because you won’t have both British and American units shucking through on the same turn. In some cases you can even “trap” Allied ships in the Med if they attempt to shuck through by snagging Gib after they have moved in.
As stated, once you’ve made the move on the neutrals, Germany loses that Swedish Ore NO - but at that point you’ve more than made up for it in IPC resources by securing Russia - making it a semi moot point.
All in all for the Axis, the neutral move appears to me to be a late game move to hinder Allied support towards Cairo.
Granted, Italy can simply fly SBR missions from S.France to keep the NB inactive for American use - but ultimately as the Axis you want to control Gib in order to protect Rome from Allied invasions. The neutral move is basically the most efficient way to accomplish both objectives of denying the NB and protecting Rome.