What is the difference between the two?
Why and where is the latter very strange sounding variant used?
"I want" is a statement of the fact that you desire something.
"I am wanting" is a statement of the fact that you are lacking something. "Wanting" here has the meaning of 'deficient in some part, thing, or respect'
"I am wanting" sounds very strange because "wanting" used in this way is a little dated, or a little archaic. People generally no longer uses "wanting" in this manner.
In fairly rare, formal English, "to be wanting" can mean "to be lacking", "to be deficient":
Several aspects of his performance were sadly wanting.
Apart from that, "want" with its usual meaning is normally a stative verb: in other words, it suggests a "general property" rather than the types of 'punctual' meaning that would usually warrant progressive tenses.
But, like other 'stative' verbs such as "love", "like", "know" etc, that's really just a matter of percentage usage. If the context strongly warrants it, "want" can be used in the progressive just like any other verb. You'll find common examples if there is a 'punctual' adverb such as now present and when want means something like "make a deliberate momentary effort to obtain":
"I'm really wanting to find out when it will be finished rather than how much it will cost."
(Progressive emphasises the notion of 'the reason I am asking now'.)
I arrived on the scene to see three men shouting at the official: they were wanting to know what time the ceremony would start.
(Progressive describing a 'general background' to other punctual events.)
'Wanting', in the sense - 'lacking in necessary quality', functions as an adjective as in 'the company's seriousness in trying to bring down the attrition rate has been found wanting'.
Therefore, 'I'm wanting a car' would seem to mean -'I lack a car'. More natural would be – 'I want a car'.
'Want' belongs to a category of verbs called 'stative verbs' and these verbs are normally not used in the progressive form because they show qualities not capable of change, unlike the 'dynamic verbs'. Of course, there are contexts in which some stative verbs can be used in the progressive form but I'm afraid dealing with that part right now would be out of the scope of your question.
If you are interested, here is a list of common stative verbs (verbs which are normally not used in the continuous or progressive form):