The most literal meaning of "fledge", going back to the origins of the word (from the same root as modern German "flug"), is "become ready to fly". A related meaning, almost as old, is "to grow feathers" as in its variant spelling "fletch". A bird is "fledged" when it has flight feathers; since feather growth is a gradual process, a bird may not be "fully fledged" for a little while. Take chickens, ducks and geese, examples which will have been quite familiar to 16th and 17th century Englishmen. The hatchlings are covered in fluff; when they start gaining real feathers they become fledglings. Note that the feathering process does not necessarily correlate with the timing of leaving the nest: these three types of birds all can be found outside the nest while still hatchlings.
"Juvenile" is a general term for youngster; some birders use it to refer to a particular plumage but don't always agree which one; others invent terms like "juvenal" to try to avoid ambiguous terms. "Fledgling", however, is pretty clear: it's a bird growing (or just grown) its first real feathers, before any feather molt. If it's still awkward and babyish, it's a fledgling. Whether that ends as soon as all the feathers are in, or whether there has to be a molt before a fledgling becomes something else, I couldn't say.