Grid A                    Grid B      
     A   B   C   D             A   B   C   D  
   +---+---+---+---+         +---+---+---+---+
1  |   |   |   |   |      1  |14 |23 |34 |14 |
   +---+---+---+---+         +---+---+---+---+
2  |   |   |   |   |      2  |31 |42 |26 |26 |
   +---+---+---+---+         +---+---+---+---+
3  |   |   |   |   |      3  |22 |24 |44 |29 |
   +---+---+---+---+         +---+---+---+---+
4  |   |   |   |   |      4  |12 |32 |19 |16 |
   +---+---+---+---+         +---+---+---+---+

The number in each cell of grid B is the sum of the horizontal and vertical neighbors of the corresponding cell in grid A.

NB. The letters and numbers around the edge of the grid serve no purpose for the solver. They are to be used for identifying cells in the solution.

The first thing to realise is that B1+A2=14, so C2+B3=42-14=28.  Similar pairs can be made from all four corners. This simplifies the programming, which then must also eventually test those such as A1+B2+C1 = 23, etc.

DIM used(16)

CLS

FOR b1 = 1 TO 13
a2 = 14 - b1
IF a2 <> b1 THEN
 used(a2) = 1: used(b1) = 1

FOR c1 = 1 TO 13
d2 = 14 - c1
IF d2 <> c1 THEN
IF used(d2) = 0 AND used(c1) = 0 THEN
 used(d2) = 1: used(c1) = 1

FOR b4 = 1 TO 11
a3 = 12 - b4
IF b4 <> a3 THEN
IF used(b4) = 0 AND used(a3) = 0 THEN
 used(b4) = 1: used(a3) = 1

FOR c4 = 1 TO 15
d3 = 16 - c4
IF d3 <> c4 THEN
IF used(d3) = 0 AND used(c4) = 0 THEN
 used(d3) = 1: used(c4) = 1

FOR c2 = 12 TO 16
b3 = 28 - c2
IF b3 <> c2 THEN
IF used(b3) = 0 AND used(c2) = 0 THEN
 used(b3) = 1: used(c2) = 1

FOR c3 = 1 TO 11
b2 = 12 - c3
IF b2 <> c3 THEN
IF used(b2) = 0 AND used(c3) = 0 THEN
 used(b2) = 1: used(c3) = 1

FOR a1 = 1 TO 16
IF used(a1) = 0 THEN
 used(a1) = 1

FOR d1 = 1 TO 16
IF used(d1) = 0 THEN
 used(d1) = 1

FOR a4 = 1 TO 16
IF used(a4) = 0 THEN
 used(a4) = 1

FOR d4 = 1 TO 16
IF used(d4) = 0 THEN
 used(d4) = 1

IF a1 + b2 + c1 = 23 AND b1 + c2 + d1 = 34 THEN
IF a1 + b2 + a3 = 31 AND a2 + b3 + a4 = 22 THEN
IF d1 + c2 + d3 = 26 AND d2 + c3 + d4 = 29 THEN
IF a4 + b3 + c4 = 32 AND b4 + c3 + d4 = 19 THEN

PRINT "."
PRINT USING "###"; a1; b1; c1; d1
PRINT USING "###"; a2; b2; c2; d2
PRINT USING "###"; a3; b3; c3; d3
PRINT USING "###"; a4; b4; c4; d4

END IF
END IF
END IF
END IF

 used(d4) = 0
END IF
NEXT d4

 used(a4) = 0
END IF
NEXT a4

 used(d1) = 0
END IF
NEXT d1

 used(a1) = 0
END IF
NEXT a1

 used(b2) = 0: used(c3) = 0
END IF
END IF
NEXT

 used(b3) = 0: used(c2) = 0
END IF
END IF
NEXT

 used(d3) = 0: used(c4) = 0
END IF
END IF
NEXT

 used(b4) = 0: used(a3) = 0
END IF
END IF
NEXT

 used(d2) = 0: used(c1) = 0
END IF
END IF
NEXT

 used(a2) = 0: used(b1) = 0
END IF
NEXT

The program did not make use of any of the caveats as to where consecutive numbers may not appear, yet a unique solution came out, satisfying those requirements.

The solution comes out:

15 10  1  8
 4  7 16 13
 9 12  5  2
 6  3 14 11