I've thrown both and I prefer the Eagle for it's versatility. When new -> sweet spot, it is great for hyzer lines and flat releases that give you a nice little "S". That little bit of high speed turn with that reliable fade allows for.. well.. "more" in terms of line shaping right out of the box. It's capable of easy to control anhyzers that can all finish differently depending on the height you give it.
And as an eagle (x mold of course) beats in beyond the sweet spot, it loses a bit of that LSS, but maintains a lot of its HSS and turns into a phenomenal turnover disc. It doesn't straight flip, crash, and burn. It's a great gradual turn that lands flat. And it can now flip up flat from hyzer and just cruise with minimal fade.
The closer my original SEX has been getting to being downright beat, the more I have been using it for turnover lines and the less I use my River. In fact, now I only use my river if I need more glide out of the turnover or for uphill hyzer-flip bombs.
Anyway. The eagle, in my opinion, is the superior mold when it comes to versatility. It is the Roc of the fairway- it can cover everything if you are patient and play them in all stages of wear.
In my experiences the Teebird does not work all lines as well as the Eagle can. Teebirds are still really great at what they do, and what they do they do better than an Eagle can mimic, but not quite as versatile a mold.