"Leisure" time, in general, means that time allotted for leisurely activities - usually rest and relaxation.
"Free time" however, can mean many different things depending upon the context.
Free time can be leisure time, but not necessarily.
When I have some free time (from work, family, etc.), I enjoy heading to the salon for a nice pedicure.
If I get some free time on my schedule this afternoon, I will try to tackle those interest reports my boss has been hounding me about. (In this scenario, "free time" refers to time that is otherwise "unblocked" with other specific work duties.)
One friend to another: Hey, if you get any free time this weekend, let's go to that wine tasting we talked about. In other words, the free time here is usually time away from family obligations and other scheduled weekend activities.