I have flown both Qatar and Emirates business with my husband a number of times. The Q-Suites is much more private as they have doors that close, and the divider between seats goes all the way down so it is almost like a double bed if both seats are fully reclined at the same time.
We both enjoy the bar on Emirates, but if it's a flight when we are trying to maximize sleep we usually don't go back there at all. As mentioned in the previous reply, the mini cabin at the back of the A-380 is not as loud as the larger cabin, but still does not feel as peaceful and relaxing as the Q-Suites.
We have had both good and bad service on both airlines, and the last 4 flights I had on Qatar each had service varying from excellent to indifferent/bad. My last few Emirates flights were in first, so don't have recent comparison with their business class service.
I think the food is better on Qatar, but the wine and champagne is better on Emirates if that is important to you.

Overall given the choice I would likely fly Qatar for the overall experience in business if the prices/travel time were similar- if I knew I would get a Q-Suite (which if you read their forum is sometime subject to luck, even if they advertise a route as a Q-Suite route).