Congrats on your 685's. They are very nice speakers. Your right, they require power, as do most quality speakers to get the most out of them, but not as much as you may think. The trick is to provide them high current power and I would think 60-70 watts from a high current amp with high damping factor would handle the 685's extremely well and maximize their performance.
Either of the models you specified above are certainly up to the task. Take your pick. However, neither are true intregrated amps. The Outlaw RR2150 is actually a receiver and the Emotivas are separates (a preamp and a power amp). If your looking for a recommendation for a true integrated with a good phono section, my usual recommendation to take a long look at the Marantz PM8004. This is amp built in Japan with top quality parts and not afraid to post it true numbers - 70 wpc, 25 amps peak current delivery and a damping factor of greater than 100. I have heard this amp powering 5000K 4 ohm speakers and was highly impressed. Some smart shopping on the internet and the Magnolias found inside Best Buys (if available in your area) will put this outstanding integrated amp right in your budget. Hope you get other opinions. This one is mine. Good luck building your system.