With a capsulorraphy, the implant is replaced into the pre-existing pocket. The front wall of the pocket is sewn to the back wall along the plane you wish the implant to sit.
With a neosubpectoral pocket, a new pocket is dissected in front of the front wall of the capsule around the implant, stopping where you want the new inframammary fold to be. Since a portion of this dissection is behind the muscle and is a new space, it is called "neo subpectoral" or "neo retropectoral."
Both achieve the same goal. One has not been shown to be better than the other. The only thing that matters for you is that you find a surgeon very experienced at fixing your problem. Depending upon the extent of your problem and condition of your tissues, a good repair is usually possible.
I am increasingly using the neosubpectoral approach for a variety of reasons: it is faster, easier, more accurate, less painful, and looks better sooner. But if the tissue is very thin and incapable of supporting sutures and the weight of the implant, I will often reinforce the bottom of the pocket with acellular dermal matrix such as Strattice.
One thing you need to know; if your doctor tells you that you need to go smaller, heed his or her advice. Big implants with thin skin can decrease your chances of a longlasting successful repair.
By the way I think you are misusing the term "bottoming out;" the methods you have referenced are used to treat "inferior implant malposition." Bottoming out is a stretch deformity of the lower pole of the breast in which the distance between the nipple and the inframammary fold increases. The treatment for it is a breast lift. Many surgeons and patients confuse these terms.
Bottoming of implants as an unfortunate issue and a difficult one to address, especially if they are longer implants. Commonly this is the case as bottoming out happens somewhat more often with larger implants, but the patients tissue characterstics also play a role.
Both of the methods you have described try to achieve a common goal, to raise the IMF or breast crease. Neither one is perfect and 100%.
The best method to really fix a bottomed out breast is to remove the implants and close the capsule, allow everything to heal and then come back in 4-6 months to place new smaller implants. This will probably have the longer lasting results compared to the above mentioned methods. Most patients however are not willing to have their implants removed.
Correcting bottoming-out (methods and surgeon selection)
In short neither method has been shown to be more effective at correcting bottomed-out implants. Both attempt to overcome gravity and the pre-existing situation. The neo-subpectoral method typically creates a new space just superficial to the prior expanded lower pole space and sutures the old one down. The capsulorraphy is a more widely done method in which the normal scar capsule around the implant is sutured to itself and the chest wall fibrous tissues to re-define the desired pocket contours. Neither are completely effective.
Other considerations would include smaller or no implants, mastopexy / breast lift, textured surface implants, acellular dermal matrix, post-op external taping or support bra, post-op activity limitations, and possibly others.
Your focus should be on selecting an experienced surgeon for the breast revision surgery, rather than pre-selecting a method for the re-operation. The entire professional service you receive is more important than individual steps of an operation. Search for quality, not features.
Bottoming out presumes that the implants have displaced too low on the chest in relationship to your breast tissue and requires repositioning of the implant higher. This may or may not be the case and other causes can mimic this clinical situation including poor choice of implant size or poor dissection of the pocket originally.
If you do need to place the implant higher this requires reproducing a new inframammary fold. This can be done by opening up the capsule and suturing the bottom margin or by dissection a new pocket over the outside of the old capsule. Each technique has its own advantages and disadvantages and I would suggest the you rely on the judgment of the surgeon you choose. The correct diagnosis to your problem is the first step to finding the correct solution.