I have used slicks a lot on motorcycles & cars, and a little on bicycles (2 sets is all). I won't bother talking motor vehicles because I don't find it very relevant.

Bicycles on smooth pavement don't need tread. I 100% agree with the Sheldon Brown article.

Where they do need tread, however, is in the dirt. Despite reasonable efforts, I find myself on dirt paths or groomed gravel roads from anywhere between 50 yards and 2 miles per week on my road bike. 23-25mm road tires are not great in gravel no matter what, but slicks take it to another level of sketchy. Conversely, lightly treaded tires like my current Conti GP4Season do fairly well as long as you don't take corners too fast or do any serious climbs or descents.

A slick should have the least rolling resistance, but I've noticed that they often make more rolling noise than lightly-treaded road tires. (A lot less noise than a 32-35mm hybrid/cross tire though.)

Slick vs tread is irrelevant in the rain, that's for sure. Traction is ALL about tread compound and what you're riding over. Oil, painted stripes and other road debris can put you down even on tires that work well in the rain.

Slick vs tread also is irrelevant with respect to flats; that's all about the puncture belts (or lack thereof) and carcass thickness. There are very durable slicks and very fragile treaded tires, and vice versa.