There are many differences but in short, a Rebbe listens to you from your soul and gives you guidance and answers from that level. Whereas a Rabbi, listens to you from a your mouth and gives you guidance and answers from what would be written in a book.
They actually refer you to a Chabad website which says a similar thing
I think their definition of a Rebbe is lovely. Giving guidance from the soul. Sometimes a person needs someone who can listen to them and advise them in ways which may be not strictly halachic (though of course Rebbes also answer halachic questions as well).
But I'm not sure that I like their definition of a Rabbi (though in many cases it may be (unfortunately) correct).
It sounds, in other words, as if a Rabbi only listens to the words of the questioner, and then opens a book which has the answer written in it (or knows the answer without having to read the book).
I know Rabbis who answer questions like this. Actually, they are better than the Rabbis who answer the question before it has been asked, or without listening to all of the words of the questioner (I know a few Rabbis like that too). And sometimes there is no alternative but to answer the question simply based on the words of the questioner. For example, a phone or e-mail question sometimes does not allow the Rabbi to fully understand the questioner - without seeing their body language and other critical information. Sometimes (for valid reasons) people ask questions anonymously, either through a friend, or on the internet, or in any other way. In such a case all a Rabbi can do is look up the answer in a book - the better the Rabbi the better the book!
BUT (to invoke the 'No True Scotsman' argument) in my opinion a REAL Rabbi (like a good doctor) will listen not only to the mouth, but also to the heart, psychology, emotions and history of the person asking the question. I know that if I have a serious halachic question I will first go to ask a Rabbi who knows me well, and who has known me for a while.
Even fairly 'simple' questions can have different answers depending on the person asking and the unspoken information they convey. In Kashrut for example, I know that some of the people who ask me questions won't believe me if I simply tell them that something is permitted (since they are convinced it is forbidden) and if I told them to do nothing they would go to get a second opinion. So I explain to them that there are some opinions that permit this, but there are also some stringent opinions, and I try to find a simple kashering action they can perform to give them the halachic answer they need.
In hilchot Nida, for example, if I know that there have been shalom bayit problems between the couple I will be more likely to rely on a lenient opinion (if I think that will alleviate the problem) that I wouldn't necessarily use for someone else.
I may give a different answer to a Shabbat question if I know the person has only just begun keeping Shabbat, or is struggling with dilemmas of work and Shabbat I will give them a different answer than to someone who has just come back from Yeshiva for bein hazemanim.
To be a Rabbi a person must know who the person is asking the question, and know what they are really asking (because often that is not included in the question, but must be clarified - sometimes the person themselves does not know what they are really trying to ask), and the more information the Rabbi has, the better will be the quality of the answer.
Unfortunately, it is very difficult to find a Rabbi who fits this description, and it can be hard to find ways of connecting to him and getting him to know you (and vice versa). Internet answers or book answers are easier. But not always correct.