In the HSK world these two can qualify as "synonyms".
Just to be clear here, 悲惨 is really "miserable” or "tragic". Which is a bit odd since in English, these two concepts are fairly far apart. Deal with it.
悲伤 = sad, but can be pushed to include (grieved, sorrowful, etc) like every bad dictionary will tell you so that you'll never ever learn the word for sad in chinese. How fucking sad because it's true!
Well, then again 伤心 may be the word for sad, who fucking knows? Or maybe it's 悲哀。 I will do some social investigations into this 社会调查！
So, I gave my teach a 测验 today. It consisted of two questions, well, really three or four, but who's counting!
测验 翻译 英-中
1. sad （形）.... _____________
I feel sad. ..._____________
2. outstanding: （Not 特别好 or 棒）... _____________
And the responses:
1. 伤心， 我很伤心
2. (left blank)
Ok, so we have some progress!
What you need to do with a lot of teachers is ask them the "base" word they were taught means the word you are thinking of. Odd that they never care to bring these things up. When you learned chinese, and you came across some base word in the language you probably ran into somethng like this:
聪明 - clever, witty, smart
often with the emphasis on "clever"
Luckily most important base words and emotions are simply never even taught! Think of how many base words in english, basic emotions or verbs you simply were never taught or never appeared in your teaching materials. It's phenomenal. What's worse is, oftentimes you'll simply be thinking you're saying one thing for years and then it turns out, you were just a fucking idiot.
Even in n词库 now looking up 聪明 only returns "clever". I cannot believe the british have fucked up english so much for so long that I can't get a decent definition of 聪明 as smart rather than clever in fucking 2009.
Anyway, the point is, when chinese people learn english they are given an english word like "sad" and next to it is one fucking chinese word "伤心”. Not 难过， not 悲伤，悲哀，悲惨什么的，就是再简单不过了。You, on the other hand are given shit. In the later stages they won't even bother to give an english trans and if they do its sure to be either wrong, not helpful, or helpful in horrible ways. YOU need to ask your teacher, no matter how bad their english is, what they were taught some english word means, because that is how they understand it, its often the most basic word, and its most often right. Besides, you can almost never find that kind of info unless you talk to people. You may be able to go check out youngans workbooks and textbooks and next time I head to a big bookiestore I'll do some more 社会调查于它。
Anyway, so back to "sad". After much discussion, we admit that 伤心 is the basic word for 'sad.' However, 难过与悲伤也是同样的意思。 区别在程度。
难过与伤心能这样用： 我很难过，我很伤心。 悲伤不行
悲伤能这样用： 感到悲伤， 难过与伤心不能。
Additionally, 悲惨 like we imagined, does imply 'miserable, tragic,etc'.
悲哀 on the other hand implies an observer viewing your situation as "sad" in the sense of "pathetic"。
Don't get me started on what "happy" is 快乐，愉快，开行，高兴。 Maybe another time if I really hate myself.