SkyPriority is the umbrella name for most of the SkyTeam Elite+ benefits. FB Silver = SkyTeam Elite only, not +, so Silver passengers do not receive the entire host of benefits (unless due to a premium cabin, etc). However, Silver passengers do receive an extra checked bag, elite check-in at the airport, etc. (the latter of which belongs to SkyPriority). So essentially, you are receiving some of the benefits ( extra bag, priority check-in, "minimal" priority boarding) but not all of them (priority security lane, full priority boarding, priority baggage, lounge access) - and thus will not officially have "SkyPriority access."
[...]
For comparison, AFAIK - the situation at *A and OW are quite different (at least in terms of labels), thus making it more complex (and interesting) when changing between airline alliances, IMHO.
As Xandrios explains above, SkyPriority is a specific set of benefits that are given pax who meet certain requirements like status levels, cabin class, etc. This is, IMO, one of the things where SkyTeam has a leg up on the other alliances as they have adapted SkyPriority as their standard branding and product for priority services alliance-wide with the same benefits and the same requirements. Meaning that if you get SP from being a Delta SkyMiles Gold or above member, you will get the same benefits if you find yourself on a domestic Korean Air flight and with the consistent branding used, you won't have to spend any time figuring out what line is the priority line in Korean.
Of course, SP only dictates the minimum requirements and airlines are free to extend part or all of them to others as they see fit, like how FB lets Silver members use the SP check-in desks at most airports, despite that level not actually conferring the full SkyPriority benefits on the member.