Emirates being there and sucking up the demand is exactly why you wouldn't send flights there if you're Qatar Airways. Emirates also appeals to a different market.
They appeal to very similar markets.
They both heavily rely on the transit market for volume. UK <-> India/ Pakistan/ Bangladesh, UK -> Thailand, and UK <-> Australia/ NZ in particular. They both also heavily sell their 'premium' experience - both being highly mixed (middle seat in business on the Emirates 777 or one of Qatar's ancient and poorly maintained A330s anyone? (both being gradually removed before anyone complains)). Obviously there are differences. Emirates having more point to point passengers vs. connecting compared to Qatar for example.
Stansted is the closest airport to a very large and relatively well off market (North/ North East London, Essex, Cambridgeshire). Plenty of reasons they might choose to operate there.
It's all about yield and whether it would provide a decent return for them. Obviously I don't have access to Qatar's numbers, but Stansted certainly seems more viable to me than, say, a return to Cardiff where they flew for a couple of years.