Buying a new desktop.....All-in-Ones or traditional?

The thing about all-in-ones is that they either use mobile-platform components or low-end desktop platform for the small footprint and heat mitigation, so they are not as powerful as a traditional tower at the same price point. All-in-ones are essentially laptops without the portability. The trade-off is the all-in-ones have larger screens for the desktop environment vs. laptops. Similar to a laptop being permanently connected to a docking station. If something breaks, the whole unit has to be serviced, including the monitor. It has the upgradeability limitations of a laptop as well.

With a desktop, you get desktop platform components, which are always superior to its mobile counterpart dollar for dollar. The drawback here is that if you really, really want the touch-screen functionality, you'll need to buy a touch monitor separately, and they're about double the cost of a decent 1080p LED monitor. The touch screen is where a large chunk of the All-in-One price comes from.

Overall, for your purposes, either one at most price points will be fine. You will just get more for your dollar (for internal components) with the desktop platform.

All-in-ones can go suck a dick.

All the disadvantages of a laptop without the advantages.

Why anyone is compelled to buy these is beyond me.

This is why bad technology often wins over good technology, because people are fricking stupid.