Ask a stupid question and you might get a stupid answer. This question, which forms the title of a book by Princeton academic Nigel Smith, is pretty odd. Shakespeare for a start was a playwright. John Milton was a poet and political controversialist. Milton's poetry is much more magisterial than Shakespeare's: he doesn't create characters as much as argue in verse. Shakespeare's poetry is tied to moments, whereas Milton's is tied to the great dramas of Christian theology- the fall (Paradise Lost) and the person of Christ (Paradise Regained). To compare them seems to miss the point- because in a sense they were never trying to do the same thing- Shakespeare touches on great themes by sketching individual lives, Milton touches on individual lives by sketching great themes. One wrote the most natural verse ever written in English, the other wrote the most artificial (Milton's lines are often filled with amazing music, but the music is much more difficult to grasp than Shakespeare's is).
Furthermore it seems to me a little stupid even to want to compare them. Milton is generally thought of as the lesser poet: but his poetry still repays great attention. He was one of the greatest writers to have ever lived and some of his lines- 'better to reign in hell than serve in heaven' will survive as long as the English language. He was also an amazingly fecund political thinker- a republican who defended the English experiment in government without a king in the 1650s, he was an early advocate for divorce and for freedom of religion. To say he was worse than Shakespeare is a bit like saying Einstein was a less important scientist than Newton- so what? It doesn't mean that you cannot understand science without understanding relativity or that you cannot really understand English literature or history without reading Milton. Milton understood that himself writing a eulogy of Shakespeare and so did Newton, commenting that those alive today stand on the shoulders of giants. Lists that rank authors are often pernicious: the idea that there are authors who you should read- a kind of top ten or even top one or top a hundred is barmy. You should read everything with any quality.
And yet.... there is a reason this book has been produced and its not because the question is a serious question... rather the question is a means. It is a means for Smith to introduce all the ideas about Milton that academics have had over the last forty years to a general readership, smuggled amidst the idea that one could prove Milton was better than Shakespeare. It is like the virtues of an Everyman catalogue: the idea of a list is epistemic nonsence- but it is didactic sense- it helps people enter the wonderful world of literature and art to know which painters and authors to look at, then they can move on. That is the purpose of this book and of literary lists or any kind of list in general, they are not meant seriously but as aides to people entering a subject for the first time. A question like this is a crutch- before you can walk unaided it is useful, once you can understand the subject, you can throw it away.
Is Milton better than Shakespeare? For those who have read them, silly question- for those who haven't read either- start with Shakespeare and move on to the later poet.