Puccini is better. Verdi was talented theatrically, but in my opinion, his orchestration is underwhelming and can even sound kind of whimsical at times.
People here have said things like, "Verdi may have been the better composer, but my heart goes with Puccini." I find this thought kind of silly. It's sort of like saying, "I should prefer Verdi, but I prefer Puccini. But, because I should prefer Verdi, Verdi is the better composer." The composer's job is to win over your heart -- not to write music that is somehow objectively better.
Puccini's operas are sappy, and Verdi was much more prolific, but Puccini is better able to move the listener with his music.
In my opinion, Puccini's masterpiece is Madama Butterfly. It's this opera that tips the scale toward him.