hard to different both of these, but there potential implications of these two when live with someone closer we always say"住一起" when live with some unfamiliar we always say“一起住”

“住（在）一起” = “一起住” ； “睡在一起” = “一起睡” 【例子如下】： A：喂C，好久不见啊。听说你现在跟D【住（在）一起】？D她人很好的哦。 C：没有啊，我们没有【住（在）一起】哦，好遗憾的。看来我得找个机会跟她【一起住】，哈哈。你说，假设我们真的合租了，到了冬天如果我想跟她【睡在一起】的话，她会不会拒绝跟我【一起睡】啊？我冬天好怕冷的。

I think there's is no difference between "住一起" and "一起住", they mean same.

一起住=住「在」一起=live with sb. together ex. I live with my sister.=我和妹妹一起住=我和妹妹住在一起

Actually, theyve got the same meaning. you can say whichever you like. :)