R, 31 bytes

\(a,b,c,d)atan2(d-b,c-a)/pi*180

A built-in atan2 function.

Attempt This Online!

result:

> f(5,5,5,-5)
[1] -90
> f(5,-5,5,5)
[1] 90

Pascal, 68 Bytes

Pascal according to ISO standard 10206 “Extended Pascal” includes complex numbers as a built‑in data type.

function f(A,B:complex):real;begin f:=arg(B/A)*90/arg(cmplx(0,1))end

FreePascal, 71 Bytes

As of version 3.2.0 the FreePascal Compiler (FPC) does not (yet) support (all) Extended Pascal features, but the standard RTL (run‑time library) includes a custom ucomplex UCSD Pascal unit providing a record‑based complex number implementation.

uses uComplex;function f(A,B:complex):real;begin f:=180/pi*carg(B/A)end

Swift, 62 bytes

import Foundation
let f={(atan2($3,$2)-atan2($1,$0))*180/M_PI}

If you're on a Mac, you should be able to replace import Foundation with import Darwin to save 4 bytes.

C, 88 bytes

#include<math.h>
typedef double d;d g(d x,d y,d a,d b){return atan2(b-y,a-x)*180/M_PI;}

Requires compiling with GCC to take advantage of M_PI being defined in math.h as a part of GCC's built-in math constants. Try it online - since ideone doesn't use GCC (apparently), an additional few bytes are needed for enough digits of π to be accurate.

TeaScript, 28 bytes

I really should of implemented trig functions...

$.atan2(_[3]-y,z-x)*180/$.PI

Try it online input is a.x a.y b.x b.y

Explanation

$.atan2(       // Arc Tangent of...
    _[3] - y,  // 4th input - 2nd input
       z - x,  // 3rd input - 1st input
) * 180 / $.PI // Converts rad -> deg

CJam, 15 bytes

l~ma@@ma-P/180*

Thought I'll get in the CJam game as well. Try it online. Input is in form of bx by ax ay. Unfortunately, this is the shortest method of doing this challenge without copying Dennis' answer.

Simplex v.0.7, 13 bytes

I'm glad I added mathrelations :D Unfortunately, I cannot take pointwise input. So, I input each point as a separate number (Ax, Ay, Bx, By). (I used this as a resource.)

(iRi~^fR)2LSo
(       )2    ~~ repeat inner twice
 iRi          ~~ take two chars of input (x,y)
    ~         ~~ switch top 2 on stack
     ^f       ~~ apply atan2 on (y,x)
       R      ~~ go right
          L   ~~ go left
           S  ~~ subtract result
            o ~~ output as number

I can save a char if I can take input as (Ay, Ax, By, Bx):

(iRi^fR)2LSo

JavaScript, 49 bytes

(a,b)=>((c=Math.atan2)(...b)-c(...a))/Math.PI*180

Input is taken in form: [aY, aX], [bY, bX] (notice the reversed x/y)

Javascript, 66 bytes

let f=(a,b)=>(Math.atan2(b.y,b.x)-Math.atan2(a.y,a.x))*180/Math.PI;

demo

Octave, 43 bytes

f=@(a,b)(cart2pol(b)-cart2pol(a))(1)*180/pi

Input/Output:

octave:40> f([5,5],[5,-5])
ans = -90

octave:41> f([1,0],[0,1])
ans = 90

Python 2.7, 73 Bytes

from math import*
f=lambda A,B:degrees(atan2(B[1],B[0])-atan2(A[1],A[0]))

Test:

f((5,5),(5,-5)) #-90.0
f((5,-5),(5,5)) #90.0

Julia, 18 25 bytes

f(A,B)=angle(B/A)/pi*180

This assumes that "any convenient form" already allows for A and B to be given as complex numbers. Then, the complex number arithmetic does all the heavy lifting.

Edit: converted snippet to function. 18 byte version only works in the Julia REPL.

Ruby, 64, 58 bytes

a=->(b){b.map{|c|Math.atan2(*c)}.reduce(:-)*180/Math::PI}

Usage

a.call [[5, 5], [5, -5]] # => -90.0
a.call [[5, -5], [5, 5]] # => 90.0

Mathematica, 22 bytes

{-1,1.}.ArcTan@@@#/°&

Example:

In[1]:= {-1,1.}.ArcTan@@@#/°&[{{5,5},{5,-5}}]

Out[1]= -90.

In[2]:= {-1,1.}.ArcTan@@@#/°&[{{5,-5},{5,5}}]

Out[2]= 90.

Pyth, 11 bytes

.t-FPM.jMQ6

Demonstration

Input is given in the format:

[[Bx, By], [Ax, Ay]]

If it is desired that A comes first, this can be changed for 1 byte.

Explanation:

.t-FPM.jMQ6
               Implicit: Q = eval(input())
      .jMQ     Convert input pairs to complex numbers.
    PM         Take their phases (angles in the complex plane).
  -F           Take the difference.
.t        6    Convert to degrees

Minkolang 0.9, 112 bytes

I really want to implement trig functions as built-ins now...but this was fun! (Caveat: this outputs the positive angle difference, not the signed angle difference. Given my limitations, I think that's justified.)

4[n]0c2c*1c3c*+r4[2;1R]r+1R+0g*12$:;$:8[0ci2*3+d1R;0g$:1i1+[i2*1+d1+$:*]*]$+'3.141592654'25*9;$:$:12$:r-66*5**N.

Try it here.

Explanation

I'll post a fuller explanation if anyone wants it, but the gist of it is:

4[n]                                    Take in 4 integers from input
0c2c*1c3c*+                             dot product
r4[2;1R]r+1R+0g*12$:;                   magnitudes of vectors
$:                                      dot product divided by magnitudes (z)
8[0ci2*3+d1R;0g$:1i1+             *]    Taylor series for arccos
                     [i2*1+d1+$:*]      In particular, the coefficient (1/2 * 3/4 * ...)
$+                                      Add them all up!
'3.141592654'25*9;$:$:                  Divide by pi for converting to degrees
12$:r-                                  Subtract from 1/2 - I now have arccos(z)
66*5**                                  Convert to degrees
N.                                      Output as number and stop.

CJam, 14 bytes

q~::ma:-P/180*

This is a full program that reads the input as [[Ax Ay] [Bx By]] from STDIN.

Try it online in the CJam interpreter.

How it works

q~             e# Read and evaluate all input.
  ::ma         e# Replace each point (x, y) with atan2(x, y).
               e# This returns its angle with the positive y axis, measured clockwise.
      :-       e# Compute the difference of the two resulting angles.
               e# This returns the angle between the points, measured counter-clockwise.
        P/180* e# Divide by Pi and multiply by 180 to convert to degrees.

TI-BASIC, 13 bytes

For TI-83+/84+ series calculators.

Degree
Input Y
min(ΔList(R►Pθ(Ans,∟Y

To use this program, enter the list {x1,x2} through the Ans variable, and {y1,y2} at the prompt.