The (thee) apple and the (thu) carrot

The - with the long e sound is used before words starting with a vowel
The - with the short u sound is used before words starting with a consonant

http://www.englishclub.com/pronunciation/the.htm

Capable and Able

This sentence will help explain the difference in one context.

"I am capable of turning out the light, but I am not able because there is a large cabinet in the way."

Capable - you have the knowledge, strength or skill
Able - Nothing is stopping you from doing the activity

A quote from a friend who teaches English at the university level:

"Never thought of that distinction before but the sentence makes sense. So, on this account, << able >> implies both ability and possibility. << Capable >> implies ability only. BTW, it's a distinction used in ethics as well. If I know what is the right thing to do, but cannot do it ... example, not know how to fly a plane to get my injured wife to the city ... then I am not ethically at fault for not flying the plane."


Here is another context in which able is synonymous with good:
http://meghachopra.wordpress.com/2013/04/24/able-vs-capable/

and another that expands the contexts for usage:

http://www.english-for-students.com/Able-and-Capable.html