You should use simplify_full() instead of simplify():
sage: var('s t')
(s, t)
sage: thirdroot = ((s^2 - 1)*t^2 - s^2 + 1)/(s^2 + 2*s*t + t^2)
sage: factor(thirdroot + 1)
(s*t + 1)^2/(s + t)^2
sage: a = thirdroot + 1 - (s*t + 1)^2/(s+t)^2
sage: a
((s^2 - 1)*t^2 - s^2 + 1)/(s^2 + 2*s*t + t^2) - (s*t + 1)^2/(s + t)^2 + 1
sage: a.simplify_full()
0
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