편리 = convenience; coming in handy for some specific purpose.. 편리 is very widely used as an adjective (편리하다, 편리한), but less so as a noun.
편의 = convenience; general ease and nicety for a purpose or need.. 편의 is more formal and conceptual than 편리, so it is not used as much in everyday situation. It is almost exclusively used as a noun, as in 시민의 편의 or 사용자의 편의 - there's no word like 편의하다. 편의점(convenience store) means a convenience store.
- 세상이 참 편리해졌다 = The world has become so convenient. (Can't use 편의 here)
- 쓰기(에) 편리한 기구 = 쓰기 편한 기구 = equipment convenient/easy to use.
- 지하철 승객들의 편의를 위해 임시 연장 운행합니다 = We will have a temporary extension to the run schedule for the convenience of subway riders. (편리 may be used as a second choice)