i think the difference between this two words is the different prerequisite and back ground or it depends on who u are talking to... such as , u are talking to your teacher about the homework .. your teacher might say: 你可以用你的业余时间来完成你的作业。。 and a journalist might ask an athlete : 你业余的时候都在做什么？ all this "业余” come after u have a professional role , in another words, the conversation u have before must relate to work.(profession)..because “业余” is the opposite of "业（work.(profession)“ so ,it will be weird u say "业余时间“， if u don't mention the "业” before there are not much limit 空余时间，so u can use it at will

« 业余时间 » est « le temps après travail ». « 有空的时候 » est « libre, disponible ». Tous les deux sont presque pareils. « 有空的时候 » est une expression générale et peut être employé dans tous les cas. Par contre, « 业余时间 » indique le temps après travail et sous-entend le contraste entre le travail et les loisirs que tu fais après le travail. Donc si l’on te demande que « 妳业余时间喜欢做什么 ? » Tu réponds « 睡觉 ». C’est une réponse très bizzare. Parce qu’on te demande plutôt tes loisirs. Natation, yoga, danse, etc..

业余时间is confirmed in a time，ex,you work in weekday from 9:00am to 6:00pm,so the 业余时间 is weekend and 6:00pm-8:00am next day of weekday。 有空的时候:① is a break，it can happen when you are working or you are off. ex,although you are working, you are not busy,you can see 我现在有空。 although you are off on weekend you have to study Chinese,so the weekend is not availabe to shopping or hang out with friends. 周末我没空。 ②有空的时候can be short can be long。 2：00-3：00我有空。我只这半个小时有空。我整个5月都有空。or just few minutes。——现在有空吗？——现在有空。maybe i‘m busy after 5 minutes。

业余时间 ：the time except working 有空的时候：sometimes you are free or you can control

业余时间 is a written statement 有空的时候 is a oral statement