What is the difference between being instructed to land "as soon as possible" or "at nearest suitable airport"?
I would interpret them as:

ASAP means the plane is going to kill you. Soon. So land before that happens. Short runways, military airports, abandoned airfields, a decent highway, dry lakes (e.g. Edwards AFB), calm bodies of water (e.g. the Hudson River) are all possibilities. Do not concern yourself with operational issues like taking off again, and bureaucratic formalities like immigration will be summarily ignored. Obviously if you are near the Hudson you would try very, very hard to reach JFK, Teterboro or Newark, but sometimes that doesn't work out.

ANSA means try for a real airport considering immediate operational and bureaucratic requirements, but not beyond immediate. this could include turning around and landing at Goose Bay. Basically I'm going to punch "Nearest" on the Nav panel and make an educated choice. Most Nav sets will only show open airfields that can accommodate your aircraft. For example, Gimli would not have appeared for Air Canada flight 143, assuming they had a suitable unit back then and the power hadn't gone out. 

The terms are deliberately unclear to give the pilot some flexibility - you can't predict and write a procedure for everything.