(1a) They are both but, but they are different buts. -- This but corresponds to aber, you cannot use sondern here.
(1b) Jimmy went on the swings, but Johnny preferred the roundabouts. -- Likewise.
(2a) Not only were they late for the party, but they didn't bring any beer. -- This but corresponds to sondern, you cannot use aber here.
(2b) We don't want to eat in the cafe, but in a proper restaurant. -- Likewise.
The aber-but (type 1) joins two thoughts which it would be inappropriate to join with and, because they mildly contradict each other. Think of it as "but in fact".
The sondern-but (type 2) introduces a second alternative after a negated first. Think of it as "but instead".