This answer is not supported by any references (other than the one I added to the question) but is an explanation from the experience of a native English-speaking professional scientist.
In an attempt to make this answer generally comprehensible let me first clarify the context, in particular to explain what the ‘sites’ are. LDH stands for ‘layered double hydroxide’ and the layers are the outer two sheets in the image below:
They contain lots of negative charges at particular positions — sites.
Between them are molecules (terephthalate dianions) with positive charges, each interacting with a specific position (site) on the sheet. (In fact they are interacting with sites on both sheets as they have two positive charges.)
In so far as the net effect is concerned, “balancing the charges on the sites” is equivalent to neutralizing the charge on the site (making it “charge-neutralized” in the jargon used). Why the different terms?
Cause and effect. ‘Neutralization’ is the effect of the ‘charge balance’.
Emphasis or focus. ‘Neutralization’ only describes the final state of the sheets (net zero charge). ‘Charge balance’ focuses on how it is achieved — by the interaction of the positive ‘ends’ of the molecules with the negative protruding ‘balls’ (hydroxyl groups, I assume) on the sheet.
Indication of the dynamic state. The LDH sheets are static and fixed, but the wee terephthalate molecules can float about in the milieu. So although at neutrality one has ‘balance’, a situation of imbalance (e.g. net negative charge) might have existed previously, and might occur if the molecules were attracted elsewhere. The word ‘balance’ here brings to mind the oscillations of an old-fashioned double-pan balance (or a children’s see-saw).