The differences are straight forward.
Off-site SEO is geared towards building incoming links that increase page rank and traffic. Another goal of off-site work is to build a positive reputation for the target site and to establish that site as a "go to" expert or authority on the particular niche of that website.
On-site SEO focuses on producing relevant content that awaits search engine users who find the target site. Also structure and design efforts focus on creating a make sense navigation structure that is user friendly. Also the design and structure should be easily read by search engine spiders who crawl the web as an effort to organize all the web pages in a vast index that determines when and where to place a search listing for any given search query that a user submits.
Both on-site and off-site are necessary components of search engine optimization.