Why specify a basket style versus a screen style solids inceptor? 
The primary consideration is how the unit will be maintained. A basket style is considered "DIY maintenance" meaning somebody on site will remove and dump out the basket at the appropriate interval. A screen style is used where professional maintenance is coming in with a pumper truck and sucking the contents out.

So why not use a basket? Capacity. Take for example the Striem PS-35 Solids Interceptor with a 35 gallon liquid capacity. The PS-35-B basket style has 5 gallons of solids capacity whereas the PS-35-S screen style has 17.5 gallons of solids capacity. In this case the screen style is going to have 3.5x the capacity and therefore pump-out interval of the basket style.