I have both.
The primary differences boil down to formatting/UI and the fact that News+ combines the content of the old Apple News and Texture.
There are some differences in available magazines though. One that I noticed is that Texture has Smithsonian Magazine, and News+ does not. News+ does seem to have some of my Smithsonian back-issues available to me though (at least those I've downloaded previously in Texture).
I haven't confirmed this yet but I think News+ actually has more magazines than Texture, but I didn't see anything new that I was all that interested in.
My guess (personal opinion only) is that Texture will be left to wither on the vine. As publishing contracts for Texture expire they may not be renewed unless Apple convinces publishers to let them publish through both Texture and News+.