Baby = infant Babe = a really beautiful woman ;)

"Babe" CAN mean "baby," but it is old-fashioned and I would avoid using it that way. Examples of the old usage: "The shape of the little trough, the shape of the rockers beneath, the shape of the babe's cradle"--Walt Whitman "The babe sat with his feet dangling high from a precarious infant chair..."--Stephen Crane The legendary Paul Bunyan's blue ox was named "Babe," a joking reference to the fact that she was enormous. "A babe in the woods" means an innocent person in danger, a reference to a fairy tale similar to "Hansel and Gretel." A "babe in arms" is an old-fashioned expression for a baby who is too young to walk and must be carried by the mother in her arms. It is a strange fact that, in U.S. culture anyway, adults can behave in infantile ways as an expression of playful sexuality. Thus, to call an adult woman a "girl" can mean either that she is being regarded in a demeaning way or that she is being viewed sexually. In 2014, the most common meaning of the word "babe" is "an attractive woman, regarded sexually." It is also colloquial. It is verging on disrespectful--some would say simply that it is disrespectful. It should probably not be used by a non-native speaker. Some would say it should not be used by native speakers, either.

Baby could also mean "Honey" or "Sweet Heart", like a nickname for you loved ones.

Baby = infant ... and can be used instead of 'Sweetheart' etc. Babe = 'sweetheart' ... and can be used for infants, especially in poetry etc. Examples: The child was just a babe in arms (a baby); Hey! Babe! What're you doing tonight? (said to a young woman);

Let's be academic and look at a dictionary. This is the definition of "babe" from the Oxford Dictionary: "Pronunciation: /beɪb/    1. (literary) A baby; 2. [also babes] (informal) An affectionate form of address, typically for someone with whom one has a sexual or romantic relationship." Let's look at an example each. For 1, a very famous passage in which "babes" is used is Psalm 8. Psalm 8, King James Version (KJV) 1 O Lord, our Lord, how excellent is thy name in all the earth! who hast set thy glory above the heavens. 2 Out of the mouth of babes and sucklings hast thou ordained strength because of thine enemies, that thou mightest still the enemy and the avenger. 3 When I consider thy heavens, the work of thy fingers, the moon and the stars, which thou hast ordained; 4 What is man, that thou art mindful of him? and the son of man, that thou visitest him? 5 For thou hast made him a little lower than the angels, and hast crowned him with glory and honour. 6 Thou madest him to have dominion over the works of thy hands; thou hast put all things under his feet: 7 All sheep and oxen, yea, and the beasts of the field; 8 The fowl of the air, and the fish of the sea, and whatsoever passeth through the paths of the seas. 9 O Lord our Lord, how excellent is thy name in all the earth! -END- For 2, there is a page on Pinterest, devoted to photos "hot babes" for example, which has nothing to do with infants in the summer.