These conjunctions all could translate to “and” in English, but they have stylistic differences and different usages.
Gēn (跟) is usually used in colloquial expressions, jí (及), yǔ (與), tóng (同) and bìng (並) are more used in writing language. Hé (和) can be flexibly used in conversation and writing.
Concerning functions, tóng (同) and gēn (跟) can connect pronouns, noun or noun phrases, and the items connected by tóng (同) and gēn (跟) can be transposed without changing the semantic meaning of the expression.
wǒ tóng /gēn tā shì xiào yǒu.
He and I are alumni.
Jí (及) is used for joining nouns and noun phrases, and it has to be placed before the last item when more than two items are being connected in the sentence. Semantically, the juxtaposed elements connected by jí (及) could be not equally important; the items before jí (及) is given more emphasis than the ones after jí (及) sometimes.
tā huì yǎn zòu gāng qín 、xiǎo tí qín jí qí tā jǐ zhǒng yuè qì。
He can play piano, violin and several other instruments.
Yǔ (與) can be used for connecting nouns and predicates, including verbs and adjectives.
zài dà zì rán miàn qián, rén lèi yǒu shí shì nà me de cuì ruò yǔ miǎo xiǎo.
In the face of nature, human beings are sometimes so fragile and insignificant
Hé (和) is the most flexible conjunction among all these words, which can connect juxtaposed words or phrases belonging to the same parts of speech. It can replace jí (及), yǔ (與), tóng (同) and gēn (跟) in most cases as conjunction.
kǎo shēng xū zì háng zhǔn bèi qiān bǐ tóng /gēn xiàng pí.
Candidates should prepare pencils and erasers by themselves.
zhè jiàn shì qíng hái xū yào jìn yī bù de diào chá hé fēn xī.
This matter needs to be investigated and analyzed further.
As for bìng (並), it is used to connect disyllabic verbs, and the use of bìng (並) has the semantic effect of indicating that the action of the second verb comes after, in addition to the first one.
wǒ zàn tóng bìng jiē shòu zhè gè jué ding.
I agree and accept the decision.
Unlike English conjunctions, none of these conjunctions can be used to connect sentences or clauses besides except for bìng (並).
tā zhǎo dào le zhè bù jī qì de máo bìng, bìng yán jiū chū le xiū lǐ de bàn fǎ.
He found out what was wrong with the machine and worked out a way to repair it.