Internal impression vs visual evidence
好像 is your general sense that something is the case, based on vague signals or intuition. 这道题好像不对 (this problem seems wrong) is a judgment you arrived at thinking about it. 看起来 is literally "look up and come", meaning "by visual appearance": 这道题看起来不对 (this problem looks wrong) based on what you see on the page. The distinction matters when the evidence source is specific.
他好像不太高兴,但看起来还好。
Tā hǎoxiàng bú tài gāoxìng, dàn kànqǐlái hái hǎo.
He seems not very happy, but he looks okay.
好像 works with any sense evidence
You can use 好像 when the evidence is smell, sound, touch, or pure guess: 好像有人来了 (sounds like someone is coming), 好像下雨了 (seems to have rained, hearing it). 看起来 is visual only. If you hear a noise outside, say 好像有人在敲门 (sounds like someone is knocking), not 看起来有人在敲门 (which implies you can see them, which defeats the point).
看起来 expansion family: 听起来, 闻起来, 尝起来
The V + 起来 pattern extends to other senses. 听起来 (sounds like): 这个主意听起来不错 (this idea sounds good). 闻起来 (smells like): 花闻起来很香 (the flowers smell fragrant). 尝起来 (tastes like): 这道菜尝起来辣 (this dish tastes spicy). 摸起来 (feels like to the touch): 丝绸摸起来很滑 (silk feels smooth). 好像 covers all of these meanings generically; the 起来 family specifies the sense.
这道菜尝起来很咸,听起来好像加多了盐。
Zhè dào cài chángqǐlái hěn xián, tīngqǐlái hǎoxiàng jiā duō le yán.
This dish tastes salty; it sounds like too much salt was added.
好像 as softener
好像 softens assertions. 我好像迟到了 (I seem to be late) is more polite than 我迟到了 (I am late). It hedges the claim, giving the listener room to disagree. In politeness speech, 好像 is used constantly to avoid sounding too direct. 看起来 can also soften but less flexibly; it ties the softening to visual appearance.
Position: both come before the adjective/verb
好像 goes before the adjective or verb: 她好像病了 (she seems sick). 看起来 also goes before: 她看起来病了 (she looks sick). Both accept subjects before them and other modifiers after. Unlike English "seems" which can take infinitive complements, Chinese 好像/看起来 just take a clause directly.