Where to stay—Shinjuku vs Aoyama?

I'm planning a 4 day trip to Tokyo and overwhelmed with deciding where to stay. I think I've narrowed it down to either Shinjuku (Kimpton or Hilton) or Aoyama (Aoyama Grand or Allamanda). I'm hoping to get some pros and cons of each, reviews of the hotels or suggestions for others.

Here are some other city's neighborhoods that I've stayed in and loved, so similar vibes would be great:
Barcelona - Gothic Quarter
Berlin - Mitte
Chicago - Fulton Market
Rome - San Lorenzo

UPDATE: Thanks for all your help! I’ve decided on Aoyama and trying to book the Aoyama Grand but they don’t accept Visa cards through their website. They sent me a link to pay via PayPal. I’ve never booked a hotel through an emailed PayPal link before… is this legit??