Several points to make.

First sin(angle) and cos(angle) expect angle to be in radians NOT degrees. The initial values you set are obvious degrees.

When rotating in the east<>west direction the angle can be in the range 0-360 degrees (0 - TWO_PI radians) and the rangge for north<>south is 0-180 degrees (0 - PI radians). In your program the range of values for N<>S is 360 rather than 180 degrees.
 
Also you have disabled the depth test which reinforces the problem.

In the code below I have enabled the depth test, change the initial values for the angles to radians, changed the backgorund colour to make it easier to see what is happening and added a new function fixAnle to keep the angles within the permitted range.

I have also removed the redraw command from the updateCamera function since this method is called from the draw() method so is not needed. Also it is not a good idea to call redraw while draw() is being executed.

HTH

1. import processing.opengl.*;
2. PVector cam;
3. float camRadius, camTheta, camPhi;
4. private final static float compensation = 0.05f;
5. 
   
6. void setup() {
7.   size(500, 500, OPENGL);
8.   hint(ENABLE_OPENGL_4X_SMOOTH);
9.   hint(ENABLE_DEPTH_TEST);
10.   cam = new PVector();
11.   camRadius = 250.0f;
12.   camTheta = 1.22f;
13.   camPhi = 2.7f;
14.   updateCamera();
15. }
16. 
    
17. void draw() {
18.   background(64);
19.   lights();
20.   if (mousePressed) {
21.     camTheta += (mouseX-pmouseX)*compensation;
22.     camTheta = fixAngle(camTheta, TWO_PI);
23.     camPhi += (mouseY-pmouseY)*compensation;
24.     camPhi = fixAngle(camPhi, PI);
25.     println(camPhi);
26.   }
27.   updateCamera();
28.   camera(cam.x, cam.y, cam.z, 0, 0, 0, 0, 1, 0);
29.   sphereDetail(15);
30.   fill(255);
31.   stroke(255);
32.   sphere(25);
33. }
34. public float fixAngle(float ang, float range) {
35.   if (ang < 0)
36.     ang += range;
37.   else if (ang > range)
38.     ang -= range;
39.   return ang;
40. }
41. public void updateCamera() {
42.   cam.x = camRadius * sin(camTheta)*sin(camPhi);
43.   cam.y = camRadius * -cos(camPhi);
44.   cam.z = camRadius * -cos(camTheta)*sin(camPhi);
45. }