As the others have explained, we usually use “won’t” in conversation. The difference in sound between “won’t” and “want” is significant, so you can certainly learn it. If I were to attempt to spell the American pronunciation using something like Spanish orthography, it would be something like “uount” for “won’t” and “uant” for “want”. How could we possibly confuse OU and A?
Now in RP (so-called standard British pronunciation), the vowel in “want” is a bit rounded and a tiny bit backed, so that it might sound a tiny little bit like a Spanish O to you, but it’s not a real O by any means. Also the O in “won’t” will still be a diphthong as in American English (but more or less fronted than in American English) — again, the sounds are totally different. (RP has a 4-way distinction between the broad-A of “barn”, the rounded A of “Bonn”, the open O of “born”, and the closed O (actually a diphthong) of “bone”. Luckily, you don't have to worry about that. In American, the vowel in “Bonn” is the same as in “barn”, so the difference is the R sound. (Note: Some US east-coast accents have the 4-way distinction as in RP, but this is not really necessary for American English. And speakers of General American and RP can easily understand each other.
It seems to me that if you go and get more listening practice, your listening skills will skyrocket quickly. If you tell us what you’re currently doing for listening practice, we may be able to makes some suggestions.