Well, in this context, you use a comma before "which" and a full stop before "that". It felt a little weird to me, which is why I changed it. - correct It felt a little weird to me, that is why I changed it. - incorrect (This looks like you can't express your thoughts clearly.) It felt a little weird to me. That is why I changed it. - correct It felt a little weird to me. Which is why I changed it. - incorrect (This is fragmentary; it looks like you finished your sentence before you finished your thought.)

*that is* is much more definite, however, they are synonymous. Not much difference.... so yes, both can be used.

That is why and which is why can be similar in meaning but function in different ways in a sentence. In that is why, that is a demonstrative pronoun. In which is why, which is a relative pronoun. That in that is why is usually the subject of a sentence or a clause that can stand alone: I want to be involved in town government, and that is why I'm running for mayor. My mother always gives me good advice, so that is why I need to ask her opinion. His last movie was terrible, and I think that is why his career has stalled in recent years. Which is why is used to introduce a subordinate clause (one that does not form a sentence by itself): Motorcycles are dangerous, which is why we should wear helmets. The company went bankrupt, which is why their new building is unfinished. The sunsets here are beautiful, which is why many artists come to paint them. Credit: http://www.learnersdictionary.com/qa/that-is-why-and-which-is-why