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

### 世间 ###  
**the world around people**

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 literary).   
`世间所有的相遇，都是久别重逢` (feels odd with `世界`)

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### 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