Probability vs tentativeness
他可能不来 (he probably will not come) is an estimate that leans toward "likely not coming". 他也许不来 (he perhaps will not come) is a lighter speculation, "who knows, maybe not". 可能 carries more weight, like "there is a real chance"; 也许 carries less, like "one possibility among many". In probability language, 可能 is around 50-70% certainty; 也许 is 20-50%.
他可能会迟到,也许会早来,谁知道呢。
Tā kěnéng huì chídào, yěxǔ huì zǎo lái, shéi zhīdào ne.
He will probably be late, perhaps he will come early, who knows.
可能 is a noun too
可能 doubles as a noun meaning "possibility": 这种可能 (this possibility), 有可能 (there is a possibility), 不可能 (impossible). 也许 cannot be a noun; 也许 only works as an adverb. 不可能 is a complete sentence element meaning "not possible"; 不也许 is ungrammatical. For "impossibility", only 可能 constructions work.
这种可能性很大,也许明年就会发生。
Zhè zhǒng kěnéngxìng hěn dà, yěxǔ míngnián jiù huì fāshēng.
This possibility is high; perhaps it will happen next year.
可能 pairs with 会
可能 frequently pairs with the modal 会: 可能会下雨 (it might rain). This construction is standard for future possibility. 也许 can also pair with 会: 也许会下雨 (perhaps it will rain), but 也许 alone works: 也许下雨 (maybe raining). 可能 almost always needs 会 for future predictions; without 会 it pivots toward present capability meaning ("able to").
Position flexibility
可能 typically goes after the subject, before the verb: 他可能不来. 也许 has more position flexibility; it can start a sentence: 也许他不来 (perhaps he will not come). 可能 at the start of a sentence is possible but sounds emphatic or formal. For everyday speech, keep both after the subject as the default.
The cousin 或许
或许 (huòxǔ, perhaps) is a near-synonym of 也许, slightly more literary and formal. It appears in essays and polite writing. In casual speech, 也许 is the default. 或许 and 也许 swap cleanly without meaning change; register is the only real difference.