In the Pimsleur Mandarin course it uses 如果 (rúguǒ) to mean 'if', but the dictionary shows 要是 (yàoshì) to have a similar meaning:
如果: if; in case; in the event that
要是: if
Are there any differences between the meanings of the two, and how should I choose which to use?
In my opinion, they mean the same:"if". "要是" is a little informal and more common in oral speaking, while "如果" is formal, I think, but you will see both of them in books.
I could only catch up with one differences in use between them. For "要是“, because there is a "是" inside, when the verb in the "if-clause" is "是(to be)", "是" can be ommitted, however, "是" can't be ommitted when using "如果”.
Some examples:
要是我，[我]就不会那么做。 If [it were] me, [I] won't do that way.
如果是我，[我]就不会那么做。If [it were] me, [I] won't do that way.
Also, I don't agree with Tom Au that only "要是" has a "wish" mood. I think "如果" can be used in this case too.
要是我有100万就好了。 It's good if I have 1 million,really means I wish I could have 1 million
如果我有100万就好了。 It's good if I have 1 million,really means I wish I could have 1 million
The difference is quite slight. The word '要是' is, typically, an oral expression of 'if', it's fine to use it in usual talk, but might be improper to use it in any formal text.
To me the biggest difference is "flavor". Sometimes I like the way one sounds, sometimes I like the way the other sounds. For example, I think if I were giving someone an ultimatum, I might use 如果 because to me in that situation it sounds stronger and more consequential, whereas 要是 might feel weaker or more matter-of-fact (if you do this, such-and-such will happen). This might just be my own personal preference - I'm not sure if other people feel the same way or not.
The following graphics show some differences in usage of 如果 and 要是:
There is paper with 20 pages which described the difference between 如果 and 要是, but it is in Chinese. I don't have the time to translate it now. Must edit later.
要是 might be translated, if, supposing. There is a "wish" element to this construction.
如果 also means, "if" (in case) as the OP postulated, in a strictly factual sense.
我要是知道，我早知道你了。 If I knew, I would have told you.
我如果知道，我会告诉你。If I know (in the future), I will tell you.
Why do you make the problem so complex? if you just learn Chinese to Communicate in life,you need not differentiate them completely.They almost have the same meaning in spoken language,you can use both of them in spoken language. Certainly,“如果” is more official than "要是".If you write an article or report,you should use "如果".
