I did some investigation because I was curious as well.

I think one distinction is in the lowercase 'm'. Primarily around the left most stroke of the letter where the first arch of the 'm' meets. Neue Haas Grotesk's 'm' has a narrower starting stroke on the arch when it's leaving the first stroke. Whereas Helvetica has a mostly even width arch on the 'm'. This one I'm relatively uncertain on and it could be my monitor but it does seem like there's a mild difference in whatever that nook/cranny is called.

Another spot where I think the differences are apparent are with the lowercase 't'. Helvetica has an almost right angle change in direction on the inside of the base of the 't' whereas Neue Haas is smoother.

I would also look at the lowercase 'a' for differences if you really wanted to see it, Neue's is much curvier on the "belly" of the a.

Sorry if these aren't the right words, not entirely sure what to call the "parts" of a character in a font.

The difference in shape between specific characters is one thing, though it's just as important to took at the typeface in use in body text, at different sizes, different line length, and so on. The fact that people here compare typography to winetasting instead of acknowledging the fact that it's a serious and intricate skill is honestly baffling and rude. Anyway, take a look at the "gestalt" example here, where you can see how Helvetica is much wider, has bigger spacing between characters and looks almost square in comparison to Neue Haas Grotesk. The rest of the article is very interesting as well: http://www.christianschwartz.com/haasgrotesk.shtml

There are differences between all the different remakes of Helvetica. Neue Haas Grotesk just happens to be one of the good ones, and even better, you can get it for free from Microsoft for your own use, it's part of one of the downloadable font packs from them.

I would add to that though, where something is being displayed matters too. If you're displaying on screen and you're not using anti-aliasing (I don't) then whoever did a better job hinting will display better on the screen.