Difference between "correre" and "fuggire" are understandable. First is "run" and second more like "run away" as I suppose. But what about "fuggire" and "scappare" in dictionaries they have almost same meaning.
The verbs fuggire and scappare mean quite the same in several cases. They have different origins: fuggire comes directly from Latin fugio, whereas scappare comes from excappare, that is, “to free oneself from a cappa”.
The verb is also in French and English (échapper and to escape).
However, the latter has uses that fuggire hasn't. If you look at the entry in the De Mauro dictionary, several cases for scappare cannot be substituted with fuggire.
In order to be able to use fuggire, the subject should be a person or something thought of as having will: il tempo fugge (not scappa), but one can't say “mi fugge la pipì”. In other cases scappare can be substituted with sfuggire: mi è sfuggito/scappato un errore di stampa, but not fuggito.
When the two verbs are interchangeable, fuggire might be perceived as “higher usage” than scappare: thinking to my school years, I believe I wouldn't use scappare in a “tema” (written paper) if fuggire could be used.
It might help to think about the words they are most commonly used with:
- "fuggire/scappare da" + something bad
- "sfuggire/scampare a" + something bad
That's a simplification, of course, you might also see "fuggire/scappare a + better place".
Essentially, "fuggire and "scappare" imply that you actively did something to get away from something bad whereas "sfuggire" and "scampare" can be used to say you avoided a bad things by luck (even without realizing it).
- sfuggire all'arresto = avoid being arrested (usually meant as "getting away")
- scampare al massacro = survive to/avoid a massacre (also by being somewhere else)
"Correre" is quite neutral, you can translate it with "run" practically any time.