The two ENVI versions primarily differ in the interface. Both are available with the standard installation. The newer version is intended to be more like ArcPro - a single view window, vs the classic view with 3 windows: overview, 1:1, and zoom (typically 4 x). The Band Selection window used in ENVI Classic disappeared.
One of the larger substantive changes is the change in the way Regions of Interest (ROIs) are done - the file format changed for the saved ROIs.
A number of standard techniques that were useful in Classic ENVI seem to have gone away - e.g. in running classifiers, the ability to cross-train on different data sets. If you want to move between IDL & ENVI, it's easier to do in ENVI Classic.
As a practical matter, I find myself toggling between the two versions, but I've been using 'Classic' for a very long time.
Yes there are lot of differences including mosaic ( seamless in newer version). Newer version is user friendly when it comes to use tools developed by you or added afterwards to the ENVI. Dragging the image will work in newer version which is very helpful for file like .HDR, .He5.
I am using ENVI from last 5 years. For viewing purpose Classic version is more user friendly (assigning RGB, zooming, Spear tools). Both versions have their pros and cons as it depends on the user's needs.