Helvetica Neue Vs Neue Helvetica

I have a client who uses Helvetica Neue as their corporate font. They're having a few issues between Windows and Mac. When looking into the fonts they're using, I cannot find Helvetica Neue anywhere! The only family available to license is called Neue Helvetica which seems to have been around since 1957.

The other matter at hand is they need the font to work across Windows and Mac. I keep finding contrasting suggestions that OTF will work better than TTF.

So two questions:

Does Helvetica Neue exist or do I need to suggest they switch to Neue Helvetica, and if they license Neue Helvetica which type format is going to work best across Windows and Mac?