I have to agree with Mark. The Philharmonitor at $850 is probably the best among the 3 bookshelf speakers I mentioned.
I haven't heard the Audioengine 5+, but at $400/pair (including the built-in amp) it won't be close to any of the speakers I linked. The Audioengine may be good compared to other computer speakers, but its not in the same class as those others.
To answer your head-to-head sound quality question directly, I'd bet the mortgage that you would notice a large difference between the Audioengine 5+ and any of the others within seconds. And I'm not a gambling man. I can't say what you might prefer to buy, but I'm pretty certain you would hear a large difference in sound quality.
The amps in the powered speakers may not provide enough power to drive them to adequate listening levels when you are sitting farther away than at a computer desk. I found little useful information about them at the manufacturer's web site.
I'd also point out that a separate receiver like the Yamaha or HK will provide a remote control and much more flexible control over the sound. Using powered speakers without a receiver will require you to use the volume knob on the back. That may be OK while sitting at a computer, but over time in a larger room, you won't like that.
It's up to you how much money you want to spend, but if it were my choice, I'd spend most of my money on good speakers.