Suburbs are small towns outside the city, and the people who live there often come to the city to work. "Outskirts" refers to any area at the edge of a place. It could be in the city or a little outside of it. For example, Pluto is on the outskirts of our solar system -- it's not outside of the solar system, just at the edge. Another difference is that "suburb" means a small town but "outskirts" could mean a place where people don't live, such as an area with lots of factories.

From the Latin "sub" (under) and "urbs" (city), we derive the word suburbs, which is loosely understood to be a less densely populated, less built-up, area containing mostly residences. Suburbs do usually lie on the edges of larger cities. The edge of any populated area, whether large city or village, could be referred to as the "outskirts". A suburb might be an incorporated city itself, or not.

Suburbs = sub-urban areas. This is any area outside of the city centre/CBD (central business district). You can have inner suburbs and outer suburbs. Outskirts = the outermost edges of a city or town. You could use "outer suburbs" to mean the same thing, but definitely not "inner suburbs".