The closer you read, the more you see. "Spring," like most of Shakespeare's work, has layers of meaning that make even a short song like this one a rich reading experience. This layering is accomplished with, among other things, wordplay like the punning that goes on with the whole cuckoo-cuckold thing. This punning is also one of the sources of the nuanced humor that pops up in Shakespeare. Back in the day, this guy was always good for a laugh.
Iambs and rhyme are other Shakespeare standbys. Yes, these elements are part of formal poetry even today, but the Bard took it to a whole other level.
Remember that whole play-within-a-play thing? (If you just yelled "No!" calm down and see "In a Nutshell.") Well, Shakespeare wasn't afraid to complicate things this way. Take a look at Hamlet or Midsummer's Night's Dream for more meta-action.