Hi, I made the same test (in SPSS version 30 ) as you to figure out and I got these p-values:   p-value of 0.041 (one-tailed) and p-value 0.021 (two-tailed). My guess is that the reason why you don't get the same p-value in the table you look at, is because SPSS use exact tests instead of asymptotic tests (where exakt test is the best to use for small materials).  

Regarding one-tailed vs. two-tailed:

• A one-tailed test looks for a difference in one direction only (e.g., testing specifically if c > b).

• A two-tailed test checks for any difference (b ≠ c), regardless of direction.
  
  /Gunilla

Hello Cliff.  You have found the probability (not odds) that X = 5.  But p-values are tail probabilities.  The two-tailed p-value CROSSTABS is showing you is  p(x <= 5) + p(x >= 15) = 0.041389.  Here is a little demo you can play around with, if you like.  

NEW FILE.
DATA LIST FREE / X (F5.0).
BEGIN DATA
0 1 2 3 4 5 6 7 8 9 10 
11 12 13 14 15 16 17 18 19 20.
END DATA.    
DATASET NAME ds2.
COMPUTE pX = PDF.BINOM(X,20,.5).
COMPUTE LeftTail = CDF.BINOM(X,20,.5).
COMPUTE RightTail = CDF.BINOM(20-X,20,.5).
IF ANY(X,5,15) p2tail = 2*MIN(LeftTail,RightTail).
FORMATS pX LeftTail RightTail p2tail (F8.6).
LIST.

Here is the output.

 
    X       pX LeftTail RightTail   p2tail 
 
    0  .000001  .000001  1.000000  . 
    1  .000019  .000020   .999999  . 
    2  .000181  .000201   .999980  . 
    3  .001087  .001288   .999799  . 
    4  .004621  .005909   .998712  . 
    5  .014786  .020695   .994091  .041389 
    6  .036964  .057659   .979305  . 
    7  .073929  .131588   .942341  . 
    8  .120134  .251722   .868412  . 
    9  .160179  .411901   .748278  . 
   10  .176197  .588099   .588099  . 
   11  .160179  .748278   .411901  . 
   12  .120134  .868412   .251722  . 
   13  .073929  .942341   .131588  . 
   14  .036964  .979305   .057659  . 
   15  .014786  .994091   .020695  .041389 
   16  .004621  .998712   .005909  . 
   17  .001087  .999799   .001288  . 
   18  .000181  .999980   .000201  . 
   19  .000019  .999999   .000020  . 
   20  .000001 1.000000   .000001  . 
 
Number of cases read:  21    Number of cases listed:  21

For confirmation, here are results from Stata's -bitesti- command.

. bitesti 20 5 .5

Binomial probability test

            N   Observed k   Expected k   Assumed p   Observed p
----------------------------------------------------------------
           20            5           10     0.50000      0.25000

  Pr(k >= 5)            = 0.994091  (one-sided test)
  Pr(k <= 5)            = 0.020695  (one-sided test)
  Pr(k <= 5 or k >= 15) = 0.041389  (two-sided test)

I hope this helps.