They might have the same meaning, but do have different usage nowadays.

世间

the world around people / in one's life

It focuses on people or human and express in an emotional way.
这世间每天有许多事发生 means lots happening around you (or in your life).

Also often used in chinese literary works (or classic style literaryliterature).
世间所有的相遇，都是久别重逢 (feels, and feels odd with 世界).
世间所有的相遇，都是久别重逢, all encounters in life are reunions after a long time.

世界

time and space (on the earth or in novel)

It appears more objective or formal and can be used in academic or formal writing.
世界上有成千上万的生物, millions of creatures living in the world
吉尼斯世界记录, Guinness World Records
魔戒发生在中土世界, The Lord of the Rings happened in Middle-earth