I think they are almost the same. 老家 is rather informal. It only exist in spoken Chinese. Similar example will be: 父亲(formal - father), 老爸(informal - dad). There is still slight difference between 故乡 and 家乡. You can always split Chinese words into characters to analyze them. 故乡. 故 means "old". 故乡 emphasizes the place WAS your home. You had good memories there. But now, you do not live there anymore. 家乡. 家 means "home". 家乡 emphasizes the home. The place is still your home. Your parents might still live there. When you go back, you still feel home. It could be the situation that you are working in another province. But you still plan to go back to your 家乡 after you finish this work.

是一个意思，使用的场合不一样 老家---多用在口语 家乡和故乡---多用于书面表达，当然也可以用在日常交流

老家用于口语。 家乡和故乡的区别： 所谓故乡，故有老的，旧的的意思，所以故乡指比如，你全家移民纽约，但是你祖辈都生活在中国，你可以说中国是我的故乡。 家乡的话，应为家是你居住的地方，你以前住在中国，但是你移民纽约了，你可以说中国的我的以前的家乡，或者说说纽约是我的家乡

老家用于口语，故乡和家乡中故乡更深