"Huir" is not just a word of the old Spanish. It is currently widely used; maybe barely a little less than "escapar". Of course "huir" and "a fugi" are synonyms. In Spanish, an escape and a leak are also "fugas", just like the Romanian "fugă" (at least with the first meaning). And, just like in English, a fugitive is a "fugitivo" in Spanish, and a "fugar" in the beautiful Romanian language.
Yes, they both mean "to run." It was explained to me that 'a fugi' is used when a person is forced to run, such as running from a lion or running because you are late to something... Where as 'a alerga' is used for a more leisure, recreational way, like running to get some exercise or running a race.