Q:

Cyrillic alphabet in math

There are a lot of variables and constants in a paper that I am writing. Is anything wrong to use Cyrillic letters for constants and Latin and Greek letters for variables? I wonder why the letters are not used in other papers when they need extra symbols.
(Image is provided by @Ryan)

$2ж+цc$ looks better than $2b'+Cc$.
I know some readers may not know the name of these letters, but this is true for some Greek letters too. What is more important: beautiful and easy to understand equations or knowing the name of the letters? Are these letters used in the academic material of the countries that their language are written in Cyrillic?  
(Personally, I do not know the name of most Cyrillic letters. But I think using them is interesting and useful.) 

A:

What is more important: beauty and easy to understand equations or
  knowing the name of the letters?

Neither. What is important is being able to understand what the variables represent.
If $b'$ exists in a paper, there's probably a good chance that $b$ exists, too, and that $b'$ is related to $b$ in some meaningful fashion. Good mathematical writing leverages natural connections between nomenclatures to expose a natural connection between concepts.

A:

For anything you write that you want other people to read, do not use Cyrillic letters if you don't know what you're doing. If there were any trend to use Cyrillic in math then the Russians would use them, and they don't. They write almost everything with Latin and Greek letters like everyone else. And they write "sin" for the sine function, "lim" for limit, "ker" for kernel, and "Gal" for Galois group. You can see this for yourself by looking at Russian wikipedia pages for math concepts with lots of formulas: Latin and Greek letters are everywhere.  For instance, find the Wikipedia page on fiber bundle or metric space and then scroll down in the left margin to the Russian link where it says Русский (it's alphabetical by Latin letters, so this is after the more readable link to Romanian). You may not be able to read the words on the Russian pages, but the math will look "normal".
One of the few situations where Russians use Cyrillic in pure math is the abbreviation for gcd and lcm, which is НОД and НОК. Those are the first letters of the Russian terms "greatest common divisor" and "least common multiple". In applied math, Russian abbreviations are used for words (e.g., км/ч for km/h when describing kilometers per hour). That's not like variables you're asking about. The only instance I can think of where Russians commonly use Cyrillic letters as variables in pure math is when talking about a triangulation: instead of $V$, $E$, and $F$ for vertices, edges, and faces, they may use $B$, $P$, and ${\it \Gamma}$, which are the first letters of the Russian words for vertex, edge, and face. 
Some comments on the question point out the international use of Ш (esp. in number theory, after the first letter in Russian for Shafarevich). The only other specifically Cyrillic letter I have seen in international use is Л for the Lobachevsky function in hyperbolic geometry, after the first letter in Russian for Lobachevsky, though perhaps due to lack of adequate fonts it is also written as $\Lambda$. I gave a lecture in Russia last year where I was talking about Belyi polynomials (these are polynomials in $\mathbf C[t]$ whose critical values are in $\{0,1\}$) and the notation I made up was Б$(t)$, since Б is the first letter of Belyi's name in Russian. Maybe my idea will take off and in the future Б$(t)$ will be commonly used for Belyi polynomials, but please note that I wasn't just picking some random letter like Ж or Ю for the sake of using "exotic" letters without reason.
I disagree that your example $2$ж + ц$c$ looks better than $2b' + Cc$. It looks bizarre (like $2k + wc$, but worse). Using Cyrillic letters in math in this way reminds me of Feynman's story that when he first learned trigonometry he didn't like the notation sin since it looked like s times i times n. He made up his own private trigonometric notation, and everything was going fine until another student asked him a math question and he started writing out an answer in his own notations without thinking about it. The student asked "What's ${\it that}$?!?". Feynman realized that if he wanted to communicate with other people he needed to use the same notation as everyone else, so he abandoned his notation.
Finally , the Venn diagram in your question is not quite correct. In the common overlap of Latin, Greek, and Cyrillic letters are K and Y. Those are not Cyrillic letters. Take a close look: K vs. К and Y vs. У. The first is Latin/Greek and the second is Cyrillic. If you think that's not an important distinction, consider the following: a Russian math student once asked me how I was typing some math exercises in Russian because the font looked slightly weird to her. It turns out that the Cyrillic font I was using in LaTeX produced K instead of К.

A:

One already-used scheme for getting more letters, is to use boldface, script, fraktur, etc.
$$
ABC\qquad\mathcal{ABC}\qquad\mathscr{ABC}\qquad\mathbf{ABC}\qquad\mathfrak{ABC}
$$
Preferable to using symbols that your readers do not already know.

