Hi all I'm up really late had some thoughts popping around my head on what these two are trying to teach and convey, hope you don't mind me sharing. Here they are please feel free to add or comment on anything I've written.
1. To win
2. To crush the opponent
3. At any cost ( within the rules)
4. This teaches one how to dominate the opponent
5. This proves Who is physically stronger.
1. To not lose ( so no one loses).
2. To take care of the opponent
3. So there is no cost to anyone (because there really are no rules)
4. This teaches one how to create harmony.
5. This proves it's not about strength. It's about spirit.
In my experience, training BJJ, after aikido: there are some people who focus on strength, and winning, in sparring; these people can maybe overwhelm beginners - but when they go with good blue belts (the first graded belt you get in BJJ), or above, their strength is nullified: they have to use technique, if they want to 'win'.
I treat sparring as a learning tool: I have no interest in 'winning' (getting a submission) - only leanring. The great thing about sparring, is that - as opposed to aikido, where someone who gives you whatever sort of resistance, or compliance, they choose to - when someone passes your defence, or armlocks, or chokes you, you can see for yourself where you need to improve; in aikido, someone - in an entirely theoretical setting - just says 'Don't leave an opening there, or someone could do A, B, or C.'; 'losing' (as you seem to view it) leaves you in no doubt as to what you need to improve.
It seems to me that it's a lot of aikidoka who have an ego, and a fear of 'losing' - and that's why they're so threatened by sparring, and hide behind this idea of non-competitiveness.
In BJJ, there's a saying: 'Leave your ego at the door - or somebody will take it from you.' - meaning, it doesn't matter if you get 'beaten' in sparring, because you don't base your self-worth on your place in a hierarchy - the kind of hierarchy you have in, say, aikido, where people line up according to their place in a hierarchy, and there are sempai, and kohai, and you have to respect the guy who's older, and the guy who's trained longer, and the guy who...etc.
There are a lot of smiles at the end of my BJJ classes; I can't say quite the same of the aikido classes i've been to in my time.
And regards point #2 on the non-competitive list:
Aikido is a great martial art for the egotistical: they never have their delusions challenged; dealing with 'defeat' every time you practice, as in sparring/randori, forces you to be humble.