In this context, I'm free means I have time. I'm available generally means 'I'm single'. You're available for dating. I hope this helps.

"available" tends to have the connotation of "unmarried", and it could sound like the man is checking out the possibility of a potential girlfriend. "free" does not have that connotation (or at least, much less). So it just sounds like the man is asking if the woman has free time. So when the man changed from "available" to "free", he probably did so in order to sound less like he was flirting.

I agree with Jessica and Alan. I have found this item in Longman Dictionary''朗文当代高级词典“. It sais 'Someone who is available does not have a wife, BOYFRIEND etc. and therefore may want to start a new romantic relationship with someone else'. In Chinese '未婚的、无伴侣的'.

The words "free" and "available" have the same meaning.