As a term of endearment (something one CALLS their spouse / lover), "babe" is the more modern form of "baby":
"Hey, babe, do you want to do vacation in Italy this year?"
You will hear people call strangers "baby" when they are trying to gain attention (like at a club, etc.):
"Hey, baby -- you here with anyone tonight?" (I left out "are" on purpose, because this is how someone would say it at a club.)
"How you fit in them jeans, baby?" (again, poor grammar to reflect the type of street slang that would included "baby")
Those in a relationship will refer to each other as "babe" or "baby" ("baby" tends to be used by older people who grew up 30 years ago, or by people in the South of the US; rarely in the North).
Typically "babe" is used ONLY between people who are already in a relationship (i.e., dating, married).