They mostly correspond to "cute" and "pretty" respectively. In Finnish, nätti usually has the connotation of kaunis (beautiful) to at least some extent, whereas söpö is just something very cute not necessarily considered beautiful per se.
E.g. babies, small animals, and other little things are often described as söpö. Nätti then is usually used as milder or more colloquial version of kaunis, e.g. nätti tyttö (pretty girl), nätti auringonlasku (pretty sunset).
Yeah, you wouldn't. :) "Mansikki" comes from the word "mansikka" (strawberry) and "Mustikki", another traditional cow name, comes from the word "mustikka" (bilberry, blueberry). The -kki ending makes them into names, and naturally there are also people names ending in -kki, such as "Kyllikki" or "Annikki".