As Andi says, in some informal and dialectal varieties of English, 'want' is used as a dynamic rather than stative verb. In fact, even in standard English, it is common to use 'want' in some continuous tenses : while the typically Scottish form 'I'm wanting to see the doctor' would be seen as non-standard by most native speakers, a form such as 'I've been wanting to see this film for ages' is perfectly acceptable standard English.
As for your two sentences:
1. He wants to see the doctor = This is a correct present simple form
2. He want to see the doctor = This is an incorrect present simple form because the 's' is missing.
This is the only difference between your sentences!