Interesting, in German I know that phrase as "Liebe deinen Nächsten wie dich selbst", so it's more like "the one next to you" than "your neighbor". I think the phrase refers not only to spatial but (also) to social closeness, which could be the reason to use "proximulo" instead of "nejbaro" here.
Edit: Also, in German "Nächster" can also refer to social closeness, which supports my view :)
Amu vian proksimulon - Aime ton prochain
proksimulo - prochain - literaly "prochain" means next - figuratively, the person next to you - whether you're at home or not. Similar to the German notion (for that, I rely on Mr. Bohrer's intervention, as I do not speak German).
Mi invitis la najbarinon kiu vivas apud nia domo.
J'ai invité la voisine (feminine) qui vit à côté de notre maison.
najbaro - voisin - najbarino - voisine
Different words used - spuddy93 correctly suspected a difference of meaning not fully reflected in English (context permits to differentiate the meaning, but still is the same word).
In the sentence here discussed, najbarino is for a woman, which the word "neighbour" does not render alone.