就像 - just like (very similar; almost the same as)
正如 - just as (exactly the same as)
就像 is more metaphorical
Example:
他怕老婆就像老鼠怕貓一般 - He is afraid of his wife like a mouse is afraid of cats
(it is not the same because he is not afraid of his wife might kill him, but his reaction looks similar enough)
人死就像燈滅一般 - Death is just like a light going out (truly a metaphor)
正如 is more analogical
Example:
狐狸怕狼正如老鼠怕貓一般 - Foxes are as afraid of wolves as mice are of cats
(it is exactly the same kind of fear, fear of being killed)
巧克力對狗正如毒藥對人類一般有害- Chocolate is as bad for dogs as poison is for humans (clearly an analogy)
Your intuition is correct, 正如 is a stronger suggestion of two things are equal than 就像, and 就像 is more commonly used than 正如 because a similar comparison is easier to find than the exact same comparison
正 has the meaning of 'exactly' e.g. (正)是他 - it is (exactly) him; 箱中(正)是他失去的一萬元 - In the box is (exactly) the 10,000 yuan he lost.
Nonetheless, 就像 and 正如 can be used interchangeably in many instances
Example:
A nation can have only one government (就像 - just like/ 正如- just as) the sky can have only one sun
You can use two things interchangeably doesn't mean they are the same