The comparative-versus-moderating distinction
更 always asserts that the quality exceeds a previous level or another item. If you say 他更聪明, someone has to be less smart as the reference point. The reference can be explicit (比我更聪明, smarter than me) or contextual (the listener knows the prior item). 比较 does not compare against a specific other thing. It simply softens the claim toward "moderately X" or "on the higher side of X". 这道题比较难 says the problem leans toward difficult without asserting it is harder than something else.
In 比-comparisons, 更 is the amplifier
With 比 sentences, 更 strengthens the comparison: 他比我更高 (he is EVEN taller than me, implying I am already tall). Plain 他比我高 is "he is taller than me"; adding 更 intensifies the gap. 比较 cannot appear inside a 比 sentence; 他比我比较高 is ungrammatical. 比较 is used outside of 比-comparisons, as a standalone softener: 我比较忙 (I am relatively busy) needs no comparison partner.
哥哥比弟弟高,但是爸爸更高。
Gēge bǐ dìdi gāo, dànshì bàba gèng gāo.
The older brother is taller than the younger, but the father is even taller.
比较 as a hedge
比较 is how educated Mandarin speakers avoid sounding absolute. 这家餐厅比较贵 (this restaurant is on the expensive side) is less committal than 这家餐厅很贵 (this restaurant is expensive). Use 比较 when you want to leave room for disagreement or for the listener to decide. It is especially common in polite hedged speech, written reviews, and cautious professional opinions.
我觉得这个方法比较好,但是还要再讨论。
Wǒ juéde zhè ge fāngfǎ bǐjiào hǎo, dànshì hái yào zài tǎolùn.
I think this method is relatively better, but we should discuss more.
Position and structure
Both words go before the adjective: 更贵 (even more expensive), 比较贵 (relatively expensive). Both can be modified by degree adverbs like 稍微 (slightly): 稍微更好一些 (slightly even better), 稍微比较贵 (slightly on the expensive side, though this is less common). Negated, 更 becomes "not even more" rare, typically replaced with 没更: 没更好 (not any better). 比较 negates as 不太 (not too X) in most cases.