These two therapies differ fairly widely in their approach to healing - but they are both based on the concept of an emotional conflict as the starting point for trouble (whether physical or mental/emotional trouble - or likely, both). Both are used to greater and greater extent by an assortment of therapists (including, with EFT, widely in the psychology field) - but it seems that classically-trained medical personnel favor the GNM approach, as it is far more readily explained in scientific terms (even if it is revolutionary to mainstream medicos!). Both, though, can be grasped and utilized by the layperson - whether for humans or animals.
You will need to be a little creative and "leap-of-faithy" in applying these healing techniques to your pets. For one thing, there isn't much written on this angle (especially for GNM - though the way to apply this is perhaps more easily grasped than the means of using EFT on those we can't talk to). But both approaches can be "miraculous" in their effects, and I encourage you to delve into them (as I have done for years with EFT, to great and sometimes amazingly dramatic benefit).
Note that if you are pursuing a familiarity with EFT, you might also be interested in reading through the articles on other "energetic" therapies on the "Touch Therapies" page. Yes, one does touch - tap - the body with EFT; but I put it here because the emotional component is so clear, and to me, its "this leads to that" structure, as it were, resonates strongly with the premesis of GNM.
I will also note that GNM proponents painstakingly separate GNM from EFT (and any energetic therapy), even cautioning that using EFT can complicate a GNM case. While I assume that this may have proved to have been so at times, my own experience with EFT has been so beneficial that I couldn't possibly urge anyone not to do it! However, I will urge you to learn about both, and thereby see where EFT might fall short (and where it might come in very handy indeed - see the GNM page after reading about EFT). And, since both modalities involve the patient (if human, at least) as learner about himself or herself, they are more truly life-changing than any other "not psychology" therapies I can name... if you care to wrap your mind around them.
In the following pages, I give brief introductions to these two approaches, along with resources you can use to prepare to explore them...