They both mean to run, but a fugi is like the french fuire, which has the same etymology as fugitive. So when we want to say run away, we use A Fugi
According to my Romanian better half: when talking about ”to run”, there is no difference between the two terms ”a alerga” and ”a fugi” - however, if you talk about ”to run away”, you would consistently use ”a fugi”. If in doubt, think of the word ”refugee”, someone ”running away” from something.
Thanks. It's like Ukrainian words тікати (to run away), бігти (to run at the moment) and бігати (to run periodically)
An explanation given to me, by my Romanian teacher, is that 'a fugi' is more of a force to run e.g.: running around from danger, such as from an animal, or fleeing like a refugee (as SnekpitTom put it), while 'a alerga' is more so as a recreational or leisure way of running, like running for some exercise.