My question is how is the 1st sentence different from the 2nd, and what nuance does the くる add.
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Knowing there is no answer to this question for a long time, so I (new contributor) am going to answer this. These sentences, reminding me of Japanese Haiku, can be said to be equivalent in meaning, but in nuance they are not easy to find differences between them. So I would like to say that they differ very slightly in style or literary taste. According to my taste, the second is better than the first, because in these sentences beginning with the expression of 愛鳥の寝顔 followed by (、), which is very concise and impressive, I think that a concise form of 笑みがこぼれる is more natural in this context than 笑みがこぼれてくる, which sounds to me a very little redundant.
If you're asking this, your Japanese is pretty good! That said, ～てくる、in this context, means that the author is being reflective, looking at self outside in. It also implies gradual, not sudden, change in emotions. It's the difference between
(when I came to it, I realized) I found tears flowing down my face.
笑みがこぼれてくる、is like, I found myself smiling, versus I smiled.