In the first sentence "He asked me to leave early" means he directed me or ordered me or instructed me to do something i.e. leave early.
In the second case He appeared to me a good person(to be is not necessary), here appear means I observed him to be a good person, or I just thought from the appearances that he was a good person.
There are no rules here but simply the usage of two different words in different context, which you shall learn with practice.
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