Difference between 于是 and 所以
there's a blog post with examples about just this question here:

http://speakupchinese.tumblr.com/post/115029397455/suoyi-vs-yushi

To quote, it says:

所以 introduces the outcome, just as in English where we use ‘therefore’, ‘as a result’ or ‘so’.  

... Moving on to 于是. This is used to place emphasis on the chronology of two connected happenings: A –> B, in continuation. Often it’s used in recounting past events or a story, and translates best to ‘thereupon’, ‘as a result’, ‘consequently’.