The verb bàn zhe (伴著) means that one thing happens with the other, which indicates the concurrence or coexistence of the two things or situations.
lěng fēng bàn zhe dà piàn de xuě huā chuī jìn wū zǐ lǐ lái.
The cold wind blew into the house with large flakes of snow.
The preposition suí zhe (隨著) could be used to indicated that something is the condition under which something else happens. It can be understood as “following” or “in the wake of”.
shù zhī suí zhe fēng bú tíng de bǎi dòng.
The branches swayed as the wind blew ceaselessly.
suí zhe kē xué jì shù de fā zhǎn, rén lèi zhēng fú zì rán de néng lì yù lái yù qiáng.
In the wake of developments in science and technology, man has become more capable of conquering nature.
The two words can be used interchangeably sometimes, but the meaning could be slightly different.
hái zǐ men suí zhe yīn lè tiào qǐ wǔ lái.
The children danced as the music is playing.
hái zǐ men bàn zhe yīn lè tiào qǐ wǔ lái.
The children danced to the music.