Q:

Why do soccer balls have different panel patterns on them?

Why do certain soccer balls have different panel patterns on them? Is there controversy between each style? Which leagues and or tournaments use each style? 
For example, see the pictures below...

"32 panel or Telstar" - 12 Pentagons and 20 Hexagons

"Teamgeist Soccer Ball" - 14 curved panels

"Jabulani Soccer Ball" - 8 three-dimensional panels

A:

Why do certain soccer balls have different panel patterns on them? 

As technology advances, a soccer ball that holds its shape, provides true flight, and reduces air resistance is typically preferred. Many different designs (including those pictured above) strive to achieve such.
1950 World Cup Soccer Ball.
 
Soccer balls during that era were made of irregular, large, flat, and simple panels. This affected its ability to hold its shape. Soccer balls today utilize more curved, complex panels. A mathematical look on the topology of soccer balls can be found here for reference.

Is there controversy between each style? 

The "Teamgeist" soccer ball pictured above was used in the 2006 World Cup. Brazil, winners of the 2002 World Cup, had a tough time getting used to the ball. The main criticisms were that the ball was too light, favored ball-strikers over goal keepers, curved a lot in the air, and performed differently in rain. (1)

"Teamgeist" (Team Spirit) balls will produce unprecedented
  performances. -adidas, manufacturer of Teamgeist soccer balls
"I still haven't been able to find the best way to strike this ball.
  It's very light, the way they are doing it is completely different
  from before. It seems like it's made of plastic." -Roberto Carlos, defender for Brazil
"It gains a lot of velocity when it's raining." -Rogerio Ceni, goalkeeper for Brazil.

The "Jabulani" soccer ball pictured above was used in the 2010 World Cup. The criticisms were similar to the "Teamgeist." However, it's ball flight in air was unpredictable. (2) A "knuckle-ball" effect had also been reported. NASA did a study on said report. (3)

Which leagues and or tournaments use each style? 

2006 World Cup - "Teamgeist"
2010 World Cup - "Jabulani"
FIFA covers guidelines soccer balls must adhere to in their rulebook:

Law 2, The Ball
Qualities and measurements
The ball is:

Spherical
Made of leather or other suitable material
Of a circumference of not more than 70 cm (28 ins) and not less than 68 cm (27 ins)
Of a pressure equal to 0.6 - 1.1 atmosphere (600 - 1,100 g/cm2) at sea level (8.5 lbs/sq in - 15.6 lbs/sq in)

