Is there any easy example of a formal proof in mathematics and an equivalent informal proof to illustrate a difference between those two? I would really like to see some SHORT example of an informal proof in mathematics side by side with the same proof but formal one to see some clear distinctions between those two. Do you know any such a short proof or would you be able to think of one? I have browsed the internet but I have not found any.
(I am not from a math background but I just want to have an idea what the distinctions are since I am just learning to do some easy informal proofs in mathematics.)
 A: The situation is more complex than the words let on.
There is a technical concept called a formal proof. A formal proof is a sequence of purely symbolic formulas (no English words at all!) that are related to each other by certain particular rigid rules that describe which formulas can legally follow other formulas. Formal proofs tend to be very long and nigh unreadable, and most working mathematicians with PhDs and professorships would not be able to write one.
The task of a formal proof in this sense is not to convince anyone of anything -- these proofs are objects of study in the specialized field of "mathematical logic", severing as idealized mathematical models of the kind of actual proofs that mathematicians tell to each other. They're an important technical tool for exploring the limits of what mathematical reasoning can do, but nobody expects them to be used as a way of communicating mathematical insight between human beings.
Thus, pretty much every proof you will find in a textbook from grade school level up to postgraduate research is not a formal proof, except for example specimens in mathematical logic textbooks.
Now, among the usual proofs that are not "formal proofs", there are some that are called informal. Being "informal" is not a crisp category -- it's just a way to say that this proof appeals to intuition and the reader's ability to "fill in the gaps" in the reasoning, to a larger degree than other comparable proofs. That's not really a different kind of proof, but more a difference in degree -- a proof can be more or less informal.
Relatively informal proofs are used for quick idea sketches between experts who trust each other to fill in the details using shared understanding and experience. They also sometimes appear in texts at the very beginning (pre-university) level, where students are not yet expected to appreciate the level or detail that would appear in a less informal proof.
The majority of proofs in the mathematical literature are neither "formal proofs" nor particularly "informal". They're simply called proofs. 
Because "informal" to "not informal" is a sliding scale, you shouldn't hope to get a crisp example with "informal" on one side and "not informal" on the other. Oh, you'll probably get plenty of purported examples here, but mostly what you get is simply two points that are each somewhere on that scale, plus telling you which is the most informal of them.
A: You can compare two proofs of $1+1=2$ :


*

*"informal" : by definition of $2$.


*formal : derivation of the formula $1+1=2$ in Peano arithmetic.

