After establishing neutral buoyancy, your body still ascends or descends as you inhale or exhale.
To help OP with his/her initial concern: what you describe is true BUT the key is hidden in what is called inertia (I think that is the correct term - sorry English is not my first language).
Yes assuming you are perfectly neutral, once you inhale you will start assenting (and descending as you exhale from neutral), BUT there is a lag (i.e. it takes some time) between the two due to inertia. Also, your body with all the equipment etc has a big surface and hence it needs quite a bit of force to move through water. Hence from the moment you inhale (which also is not instantaneous by itself) it takes some time for the upwards momentum to build and then you start moving up noticeably. The art of breathing uw is that with some practice you will learn to "predict" the building of this momentum and counteract it (eg start exhaling at the right moment) before you even move upwards noticeably. Not only this, but you will eventually learn to use this to your favor eg to initiate and maintain slow and controllable ascents or descents with minimal effort.
The problems with new divers (occasionally myself included) is that they get neutral (or think they do) then fill in their lungs with a deep breath, then pause and think to themselves: "what is going on now?" after few moments upwards momentum builds and they think, "oh, I am moving upwards" and then "oh sh*t, maybe I am underweight", after few more moments: "I am now going upwards even faster now, I am definitely underweight" and so it goes. By that time a slow (for the time being) uncontrolled ascent has started - if they realize it on time they can still stop it, but quite often (eg if they are destructed by a nice fish, or while taking a selfie with their go-pro or dealing with an equipment issue) by the time they do it is already too late.
To solve this the diver shouldn't have started with a deep breath to begin with - just a normal breath should do. Then once the inhale is finished and once (or actually even before) you feel the upwards momentum building you should have already initiated your (slow) exhale (to counteract the building up of upwards momentum). With a bit of practice you will see that (normal) breathing won't affect much your buoyancy.
I know that all these are easier said than done, but at least that is what is going on.
I hope it helps.