There is a huge difference in what they can do clinically. There is maybe less difference in some states and only in a few specialities (like primary care). But overall, a large difference currently.
Training is vastly different. Like not even remotely similar.
Yes they have a doctorate. But if the training is really worthy of that title...there's quite some debate. Even if it is a legitimate doctorate, it's a bit misleading in a healthcare situation. Primarily due to the lay use of the word Doctor in those settings. Lawyers have a doctorate but no one calls them Doctor. So it's primarily semantics, but there's probably something to the thought that it can be harmful in Heath care.
It's unfortunate it's becoming an us vs them situation because there's value in the idea of a nurse practitioner. Some of the medical training is probably unnecessary in a lot of situations...and cost being a considerations...it makes sense to delegate some tasks. But the nurse lobbying, explosion of questionable programs, and more autonomy has led to quite the backlash from physicians.
I have a friend who is a lawyer and will always refer to himself as a doctor whenever our buddy who is a PA is around. My friend that is a PA is always throwing around fancy medical terms in regular conversation and it gets annoying so to rib him a bit the lawyer friend is always telling him to "leave that to the real doctor" or, my favorite is, when the PA has to sign a receipt and my lawyer friend grabs it and says "Wait, just let me sign it. Doesn't a real doctor have to sign off for you any way?"