Neutral grip refers to palms facing inwards - as your arms would rest naturally by your sides. Wide grip refers to flipping the hands so the movement is spread further from the body. This grip applies to almost all upper body movements and it changes the emphasis of muscles used. It’s minimal and not that important unless your a physique model - and sometimes a wide grip will feel uncomfortable for some people in certain movements (cue the bench press and all related injuries) so if a program says wide but you’re more comfortable going neutral then go for it baby. 

Here is a list of all the exercises used in these programs and the main (there are others I know) options you have for equipment. Depending on your targets certain equipment is better for certain things. Obviously you can load more onto a barbell for a back squat then you will be able to hold in a kettlebell for a squat, for example. And doing flies on cables will work your core and functionality better than prone dumbbell flies, but if size or strength is your aim then your better off on the dumbbells - you get my drift? Often you will be dictated by what equipment is available, and often you will find that you just prefer certain equipment for no particular reason. That’s great! Get comfortable and do what works for you! I love barbells, dumbbells and cables, I hate fixed resistance, bands and suspension (trx) training. It’s a preference - find yours.