on top of other people's mentioning of quality of sports video needing higher bandwidth as well of larger audiences. the not so obvious answer is the difference between twitch and espn's video streaming is in the delivery model and cost of acquiring the original content.

Twitch is more like Youtube and Netflix where they designed the technology to move the video as well as own the servers that provide the video. This gives them the flexibility to change quality of video depending on resources available and popularity of the video on a regional basis. They also keep the player experience very simple to keep the stream stable.

ESPN uses a traditional content delivery network that they do not own, and wasn't originally designed from the ground up to deliver video. ESPN's stream is also more complicated requiring features like: closed captions, complex ads, nielson ratings, as well as authentication to login for TV Everywhere and ESPNU services.

The main difference is that Twitch spend their resources designing a video delivery ecosystem that is only on the internet and people are bringing the content to them. While companies like ESPN spend their resources creating the content and uses many different mediums to deliver video: cable, satellite, over the air, and internet.

Lastly, ESPN has a significant amount of more people watching on non-internet delivery where the ads pays the bills. also their overhead costs for sport's streams is significantly higher ( look at the price of attending a game ) and require a larger revenue from a cable/satellite subscriber to offset cost. In contrast to some guy streaming a video game over the internet from his living room's xbox will not require much to start making a profit.

source: i work in video delivery industry