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I know there's a past topic with this question already, but honestly I couldn't understand the explanation very well.

Thank you very much for the input and time.

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The meanings are mostly identical. The one main difference is that 増{ふ}やす is explicitly 他動詞{たどうし} (transitive), whereas 増加する could be either 他動詞{たどうし} (transitive) or 自動詞{じどうし} (intransitive) depending on context. The intransitive version of 増{ふ}やす is 増{ふ}える.

Past there, there are some differences in connotations and register, which are inherent to the distinction between kun'yomi words and on'yomi words. There was a recent post about this subject that you might find informative:

Difference between onyomi words and kunyomi words

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First, I must say they have grammatical difference. 増やす is a transitive verb while 増加する is intransitive. Dictionaries may say it's transitive too, but I'm fully confident that the usage is already out of use. So synonyms are 増やす / 増加させる and 増える / 増加する.

In meaning, they both mean increase in numeral value/amount, but have difference in (aside from formality) that when used with objects, 増加する treats them as if they're collective, thus doesn't match with something you count one by one.

○ 固定客が増加する
?? 常連客が増加する
○ 常連客が増える
○ 常連客の数が増加する

In these examples, 固定客 and 常連客 both mean "steady customer", but the latter typically indicates those go so frequently that become some sort of acquaintances, in this case it's very difficult to use 増加.

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