AFCI wired strangely
You have an alien breaker problem

You must use same-brand breakers in your panel  (or certain very specific competitor breakers that are UL-classified for your panel, but those classified breakers are almost never stocked at retail).

Combination AFCIs detect both "series" faults (in line with the load, by listening to the wire for crackle-pop noises) and "parallel" line-to-ground arc faults (either by having a limited GFCI built in, or by listening more carefully). 

Earlier combination AFCIs used the "limited GFCI" strategy, and so, they need to be wired just like a GFCI. This also means they cannot support 240V loads or multi-wire branch circuits (unless they are built as 2-pole breakers). 

That's solved by some of the newest AFCIs, which use the "listening more carefully" strategy.  These are a technological progression from earlier design AFCIs. So I am very alarmed that your existing breakers don't need neutral, but your new breakers do.  


Maybe this is a GE panel having the fancy neutral-less AFCI breakers, and here you are coming back from the store with Eaton BR breakers because they were on sale.  No, no, no, no, no. 
Or alternatively your builder fit an Eaton BR panel, and used the new GE breakers because they play well with multi-wire branch circuits (but not BR panels), and here you are, coming back from the store with the correct BR breakers. 
Or it could be a case that you went to an Eaton dealer, and they pressed Eaton CHQ breakers into your hand for your Square D QO panel, and that combination is UL-Classififed, so A-OK.  
Or your original breakers might be the GE neutral-less AFCI, and the new breakers are the GE needs-neutral kind, e.g. because they are dual-function AFCI-GFCI, which is A-OK.  


We don't know.  

Regardless, you can't leave the uncertainty, because it's dangerous.  Let us know when you can edit your question to include more info about your panel, existing AFCIs, and candidate AFCIs, and we will backfill better info.  

Have no moral objection to returning wrong AFCIs to the store.  They should not have sold them to you.  Lots of lots of shopkeepers "conveniently believe" that any of the 1" breakers will fit any other panel, and while they will clip in, the stab does not contact properly and they will arc and set the panel on fire.