Variable Prime "Twins"

Brachylog, 27 23 bytes

:1f
hS,?tye.:S+:.L*$pL,

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Predicate 0 (main predicate)

:1f                     Find all solutions of predicate 1
                        given Input as Input,
                        Unify Output with the set of all solutions.

Predicate 1 (auxiliary predicate)

hS,?tye.:S+:.L*$pL,

hS                      the first element of Input is S,
   ?tye.                Output is an element between 0 and
                        the last element of Input,
       .:S+:.L          The list [Output+S,Output] is L,
             L*$pL      The product of L, prime-factorized, is still L

05AB1E, 9 bytes

Code:

LDpÏDI+pÏ

Uses the CP-1252 encoding. Try it online!.

Jelly, 8 7 bytes

+ÆR©_f®

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Explanation

+          add the upper bound and the difference
 ÆR        find all primes up to that number
   ©       save that in the register
    _      subtract the difference from each
     f®    remove anything not in the original prime list