Short answer, there is no real difference.
Long answer, ArcGIS Pro and ArcGIS Desktop Advanced have 2 different backends and support different… sort of business cases? Pro is kind of more on-the-fly-based and is built on 64-bit architecture. Desktop is more similar to the old ArcInfo and is more analysis-focused and is built on 32-bit architecture. The reason Desktop is still 32-bit is because they would have had to have completely re-written the entire code base (I think) to have it be 32-bit. Pro was built to basically replace it, but it actually (at least when last I used it) was not as fast or as efficient. The only downside to Desktop is that if you want to do multithreading (within a single instance of Desktop), Desktop doesn’t actually do it (it might say that it does, but it doesn’t actually). Personally, I think Desktop is more optimized, efficient, and streamlined, and has a long lineage. Pro is much newer. ESRI is of course pushing Pro. Oh, also, the licensing structure is different between the two, so you might have to check with your license administrator, if you have pooled licenses on a license server (Desktop uses pooled licenses, Pro uses named users). You can also purchase standalone licenses for Desktop for sure, not sure about Pro…) All extensions are still separate regardless if you are using Pro or Desktop. Project files (map files) are different and cannot be shared between the two (data files can, however). There’s probably other things I am forgetting, but these are some of the “big” things.
Or just use a combination of QGIS and GRASS, which QGIS seems to have all the same functionality as Desktop plus MANY (if not all) of the extensions (Spatial Analyst, 3D Analyst, Neetwork Analyst, Geostatistical Analyst) and GRASS is basically research-grade GIS (and QGIS integrates all GRASS functions as far as I can tell so you don’t even need to install it separately and learn the syntax).
ESRI software is nice if you have swam in that ecosystem for years and years (as I did) because everything is very familiar. But the licensing was frustrating, the customer support was frustrating, being in an organization that wanted to be cheap and not purchase enough licenses for both Desktop and various extensions was frustrating, getting a named license was ok but then Pro was a lot slower for doing actual analysis… But working through the QGIS tutorials I can see that, especially on the analysis side (but even the cartography side at this point), QGIS is just as good, if not better, and GRASS has always been a superior analysis platform.
(I can’t believe it is up to 10.8 already, last I used was 10.6, sigh. Maybe I should get the student license just so I don’t lose knowledge. But I don’t want to spend the money. And ESRI is like the Microsoft of the GIS world, there are other (quite good) options, they are just the behemoth.
However, if you do install Desktop, and you need help with like, anything, I can probably answer your question. I spent lots of time helping people less familiar with the software figure out how to do things (mostly just because I had the memory and knowledge of having used it for so long).
ESRI wants to be all things to all people (see all kind of other tools and stuff they do (big data tools, image analysis tools, LiDAR tools)) that can actually be done better by other companies/software… 
