显得 sounds a little more formal than 看起来, but both are fine to use in spoken and written contexts.
Their meanings though, are not completely the same. Both words can be used to express
to seem; to appear; to look. With this meaning they are interchangeable and the difference is mostly formality:
他显得很憔悴 / 他看起来很憔悴。He looks haggard.
But both words can be taken 'literally' -
(actively) present oneself as while
(passively) be seen as. When used in this way, they are not interchangeable. E.g.
这个节目看起来很好笑。<-- 显得 would sound unnatural because people find the show funny, not the show presenting itself as 'funny'.
他故意显得很愤怒。 <-- 看起来 is acceptable but 显得 is much better because it is consistent with 故意. The sentence means 'he was trying to make a point by appearing angry'.