If we look at some meanings of "screen" from the OED:
**2.b. transitive. To block or shut off (wind, the sun, etc.) by interposing an obstruction; to act as or form a barrier preventing (wind, the sun, etc.) from passing through.
Note the use of "blocking" that is not really covered by "filter"
1657 T. Watson Christs Lovelines in Saints Delight 390 How lovely is Christ who can screene off the fire of Gods wrath from thee.
2005 Age (Melbourne) (Nexis) 19 Feb. (Gardening section) 6 Planting trees and shrubs in groups also reduces maintenance because each group forms a self-supporting thicket that screens the wind.
II. To sieve, filter; to evaluate, analyse.
9. a. transitive. To obtain, remove, or separate (something, esp. impurities or unwanted material) from a substance, mixture, etc.,
1613 R. Loder Farm Accts. (1936) 50 iiij b. of seedes skrined out of my mault I value at xijs. viijd.
2010 I. Alesina & E. Lupton Exploring Materials 82 In India, cotton saris are used to filter water by screening impurities.
In the mid-20th century these old, established, uses gave rise to
11.c. transitive. To investigate or evaluate the suitability or eligibility of (a person, esp. as one of a group) for a particular purpose; to examine or check for the presence of a desirable or undesirable element or attribute.
1942 Cedar Rapids (Iowa) Gaz. 16 Oct. 18/6 USES work applicants are being carefully ‘screened’ by test to determine their abilities.
1956 W. Graham Sleeping Partner x. 82 When you said you were bringing an assistant to Harwell, of course we had to have her [security] screened.
1990 M. J. Heale Amer. Anticommunism vii. 138 Federal employees were to be screened for ‘disloyalty’, a concept that was not clearly defined.
It is this final meaning that has given rise to "call-screening".
I'm sure it is a commercial ploy to give the idea of some sort of security check. It has overtones of the "intelligence community and the military" giving the user of "call screening" the security that they think they want: A nuance that "filter" lacks.