Very nice question. And I don't quite agree with the post above. The using cases of these two words can be rather different. 还是 is used for alternative questions. For example: 你要果汁还是牛奶？ - Would you like juice or milk? And you can't use 或者 here. 你要果汁或者牛奶？ - WRONG. And the options can more than two. For example: 你的工作时间是6小时、7小时还是8小时？ - Is your work time 6 hours, 7 hours or 8 hours? What's more, 还是 can be used in statements that you don't know something. e.g. 我不确定会议上午还是下午开始。 - I'm not sure whether the conference starts before noon or in the afternoon. And you can't use 或者 here. 我不确定会议上午或者下午开始 - WRONG. On the contrary, 或者 is used for statement. For example: 我打算明天或者后天去北京。 - I plan to go to Beijing tomorrow or the day after tomorrow. 我们叫他“王师傅”或者“老王” - We call him "Master Wang" or "Wang". You can't use 还是 in these occasions. And in some cases, these two words can be interchangeable. For example, cases in which there is a "无论" or "不管" - No matter, Despite. 无论刮风或者/还是下雨，他总是按时到。 -No matter it's windy or rainy, he always arrives in time.

还是 for questions 或者 for alternative sentences (not questions)

in fact it's very simple. you must know what "subjunctive" is, right? When we want to express some emotion, we could use the subjunctive. For ur question, if it's a subjunctive, plz use 还是. Otherwise, plz use 或者. In chinese, we say: 当说话者表示疑问、假设、主观愿望、怀疑或推测时，可以用虚拟语气（subjunctive） for example, 你要苹果 还是 香蕉？（疑问） 不论刮风还是下雨，我都要去动物园。（主观愿望） for the normal case, we use 或者.

I think these two phrases are the same.