What are the advantages/disadvantages of non-standard analysis? I'm not interested in an in-depth answer. Here are some specific questions for which I couldn't find an answer:


*

*With non-standard analysis, can we solve problems that can't be solved using standard analysis?

*Do we have some results that differ from standard and non-standard fields? (E.g.: is there a function which derivative is different if calculated with the standard and the non-standard definition of derivative?)

*More importantly: is non-standard analysis just a formalism, or is it needed for practical reasons?
Simple yes/no answers are fine. Short examples are a bonus.
 A: Look at the answers to Is non-standard analysis worth learning? Do they answer your question?
If I understood your question correctly, it seems the answer is that
non-standard analysis (NSA) is not technically needed for the kind of
practical reasons you mention.
in particular, this answer to the previously-cited question
mentions a few results that were first found via non-standard analysis,
but also says these results were all found later by means of standard analysis.
Indeed it seems that theoretically, the interesting thing about NSA
is how well its theorems correspond to statements that are also true 
(and provable) in standard analysis.
On the other hand, consider this answer concerning the teaching of calculus.
It cites evidence that students of calculus learn the concepts better when things
are presented first in terms of infinitesimals and the epsilon-delta formalisms
are introduced later. Not everyone agrees with this conclusion.
(And that's an understatement.)
But I think this might qualify as some kind of practical use of NSA
if the conclusion is true.
