I just looked at Wikipedia articles about Bilinear interpolation.
A tent filter is a kind of Bilinear interpolation.
A Bilinear interpolation interpolates on 2 dimensions.
It first linearly interpolates on one dimension and then the other.
When you look at this picture, it first does it on the column and then the row for example.
The reason why this could be better than averaging colors, is that when one of the points gets closer to the center than another one, it contributes more to the final color.
With averaging the colors, all the points contributes equally, even when one point is closer to the center than another point.
If all the 4 points are on fixed positions on a equal distance from the center, then you get exactly 25% contribution on all points, that's the same as averaging.
This means that when you implement anti aliasing by just rendering at 4 times higher resolution, and then down sampling it. The output between the 2 ways should be the same, as far as I understand it.
So, which filter should you use? Well, it depends on what you want and your situation. If you pick random sub pixel locations for your samples, then it might be better to use a Bilinear interpolation.
If all the points are fixed and you just render out at a higher resolution, then averaging could be useful.
In the end this is all just some sub pixel things that normally you don't or barely notice, so if there is some kind if AA, then it's good enough. But that's just my opinion and probably that of the average viewer of 3D animation or something like that.
A Bicubic interpolation gives you a smoother look.
Which is better? Well, it depends on what you like, and I don't think that there will be a huge difference.
(Images are from Wikipedia and are public domain)
I hope this helps, I just want to say that I am not a professional on this, I'm just somebody that just likes computer graphics stuff. It could be that I made some mistakes. If so, just say it and I change it!