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simplify is a very weak thing - it only sends things to Maxima and back. full_simplify does indeed do some reasonable powerful things, of course. As far as I know, these both would just "simplify" each side separately.

To "simplify" automatically along the lines you are suggesting might be quite hard in general. If there is only one term in common among fifty on each side of the equation, and we asked Sage (or anything else) to look for that in every symbolic equation that was simplified in the "do each side" sense, one might see massive inefficiency. If there even was something to find, like how deciding whether an expression is zero is a quite hard problem.

One way to get around this is

Eq = a+b == a+c
(Eq.lhs() - Eq.rhs()).simplify()

though of course this gives b-c which isn't exactly what you're looking for.