I'm sure there have been many great Rabbonim who have written on this question. But I'd like to share some of my thoughts and limited understandings.
This time through the Tanya I think the answer is a little more clear: A Tzaddik is someone whose love for HaShem is so great that it causes him to despise sin and all evil utterly.
A Beinoni is someone who has great love for HaShem such that he is always involved in Torah and Mitzwoth, but never reaches a point such that he hates evil from the essence of his being. He must always fight against the allure of evil by mainting his great love and devotion in the service of HaShem.
As far as I can tell, the difference is in the level of commitment and effort, something that would seem applicable based on the statement of Chazal that יגעת ולא מצאת אל תאמין - never believe someone who claims he strived and did not succeed. It seems a Tzaddik serves with ever increasing devotion and investment until there is nothing left of himself but the desire for closeness with HaShem. A Beinoni on the other hand wants to serve HaShem but perhaps with some reservations or some reasons that hold him back whether by his own choice or by the influence of the world around him.
Another possible hint from the Noam Elimelech is in Parashath Naso that he states a Tzaddik who is able to truly connect to HaShem and bring down blessing on Am Yisrael is one who sees even his inclination to do evil as an outright sin that he performed of his own free will. The Baal HaTanya explains that such inclinations--momentary desires arise from the Yetzer Hara residing in (the left chamber of) the heart--when driven out of mind immediately, and not given voice even in thought, do not constitute sin at all, rather they are an indication that your Yetzer Hara has not been driven out of your heart.
There is actually an interesting issue that I never really knew about before in this context where there are two different styles of relating to one's Yetzer Hara, one's evil urge. They are enumerated in a Torah of Rav Yitzhak Meir Morgenstern here on Hayyei Sarah. The first is the level of Avraham Avinu who was able to turn his Yetzer Hara into a holy angel that helped him to serve HaShem rather than urged him to deviate from HaShem's commandments. The second level is the level of David HaMelech who drove his Yetzer Hara out of his heart entirely, and so says that his heart is an empty space. These two different levels can be united if we bring in the teaching of the Notzer Hesed which I've quoted in the past which discusses the process of dealing with the Yetzer Hara as two-staged. First we remove the Yetzer Hara, then we return only the good part of the Yetzer Hara.
It seems from my understanding of the Notzer Hesed that the Baal HaTanya and the Noam Elimelech brought here are only dealing with the first stage as delineated by the Notzer Hesed and R' Morgenstern. In this sense, we now have a fairly clear path laid out for performing the first step of the excisement of the Yetzer Hara from our heart.
This advice was more or less summed up in Rav Steinsaltz's advice I heard once in a shiur he gave on Hassiduth, that if you want to connect to Ain Sof, you have to give everything you've got, and when you've got nothing left, you're absolutely drained, you're at your end, you have to give some more.
[update: it might be coincidence that this was the 620th post published on my blog -- gematria: keter, the sefirah of ayin, nothingness, complete bitul]