Hey Bilbo, I happen to own a 2015 Monster 821 and a 2016 Scrambler Italia Independant, so I might be qualified to respond. Here are my observations:
Power: To no one's surprise the 821 has more pep, but the Scrambler is surprisingly competitive against it. When I ride head to head with my spouse, one on the 821 and the other on the Scrambler, they feel almost like equals in the power department, at least when thrashing around town at city speeds, from stop lights, etc. Regardless of what a comparison of the specs say the Scrambler feels like just as much a big boys bike as the 821 to me.
The 821 of course has different power options but I use the top 'sport' mode for all riding unless it's raining. You could totally still use the sport mode in the rain, but why not use the rain mode? The Scrambler's one power option feels good in pretty much all conditions.
Handling: The 821 is very good. The Scrambler is great. To put it another way I thought nothing could really get better than the 821 in the handling department until I got the Scrambler.
Issues: I love my 821, but it does have a few issues. Maybe 1/5th of the time it doesn't start the first time I try to turn over the engine. I let it sit for a couple of minutes and try again, and then it starts. It's never taken more than 2 try's, but it's still a little annoying. I've been told this is a problem with some duc's, and not exclusively the 821, so maybe this isn't something to sway your decision either way.
It's hard to stick the 821 into neutral when I'm at a stand still or going very slowly (e.g. Cruising to a red light). At speed it's no problem.
And here's the worst bit about the 821...the rear brake barely works. I won't go into all of the details but Ducati thinks it's a problem with the ABS unit being located inside the engine V, and they're trying to come up with a fix for me. In the mean time the rear brake needs to be bled every few weeks before it quickly becomes very mushy and then just non-existent. Appearantly this is a problem on some other Ducs as well, so like the start issue it might not be something to consider when making a decision between 2 Ducs.
On the other hand, my Scrambler has had exactly zero issues. That's just a sampling size of 1 Scrambler, but for whatever it's worth mine is game to go whenever I push the start button.
Cool factor: You decide. I love em both, and so does virtually everyone who sees them. The Monster is more iconically Ducati (though maybe in time the Scrambler will share that identity) and is, well, a Monster. There's nothing else quite like it on the road, and I admit to feeling super cool riding it. The Scrambler is pretty funky looking too though.
Value: Maybe the 821, at least from a spec sheet perspective. However, my particular Scrambler actually cost me fractionally more than my 821, and I have no regrets.
Which would I buy if I could only own one? Although I love em both, if I could only buy one I'd buy neither. Instead I'd buy the new Monster 797. It's pretty much a Scrambler made to look like a Monster and a great compromise between the two, if compromise is even a fair word to use. With that said, even with the 821 issues I've experienced I'd consider buying it again, and I'd definitely buy another Scrambler.
In response to ladybishop, I also have a 2014 Street Triple. Even if the new 765 doesn't improve on it much (and maybe it does), buy it! You will NOT be disappointed. I love all 3 bikes mentioned here. Truly you can't go wrong.
Hope that helps a bit.