Loosely translated, both "사이" and "중간" do mean "between". However, there are certain phrases/sentences that "사이" works better with, and certain ones that "중간" works better with. Specifically "사이" works well when describing three things: A) A time range 'between' two times / During a certain time 내일 3시에서 4시의 사이에 - Tomorrow between 3 and 4. B) Physical space 'between' something. 하늘과 땅 사이 - The space between the sky/heaven and earth C) A relationship 'between' two people. 걔랑 사이 좋아 - My relationship with him is good 중간 has more of a meaning of "in the middle of"/"average". The "중" in 중간 is Chinese for center/middle. So, for example: 그의 성적은 반에서 중간 정도다 - His grades were average in the class (this is directly from a dictionary). In the sentence you gave, the word 중간 matches better, as none of the usual definitions of 사이 really apply. For Koreans the choice of 중간 is just more natural; for us as second language learners we just need to hear many sentences with 중간/사이 in them and get a better feel for where these words can be best used.

do you want know meaning of "between(사이에서)" in Korean?? "between is unclear of range. a color between purple and violet. this color is not purple. and it is not violet. it's unclear color. I’ll meet you sometime between three and four o’clock. maybe they meet 3:10 or they meet 3:30 or they meet 3:40 they must meet. but meeting time is unclear. i hope you understand meaning of "사이에서" ^^