[Updated 11-12-2009]
I support a great many iPhones at work, and I love the device, but the Droid makes the iPhone look a little tired. It's not necessarily a huge leap beyond the iPhone (except for the display, which really is profoundly better, with twice the iPhone's screen resolution), it's just all around more solid.
I won't review the features here, there's plenty of that data all over the web already. I will say that the actual phone is just spectacular (you know... that one app that all smartphones are still trying to do right?) . The sound is crystal clear, and I get reception in locations my last three phones didn't. This alone makes it substantially better than the iPhone for me. (Not to mention that Verizon's network beats the stuffing out of AT&T).
If you're a heavy Google user (Apps/Voice/Gmail) then you're really going to love this phone: seamless integration with your existing Gmail contacts (not to mention facebook--half my contacts in the Droid are displaying their facebook profile picture now).
Advice: Buy the dashboard mount and car charger. The turn-by-turn GPS navigation on this thing is better than most TomToms. After you get used to asking your phone to navigate you to a destination with voice commands, or to make some calls for you, you'll find that you keep talking to it, just to see what it can do. I'm going to have to give it a name if I keep this up.
Most likely, if you're new to the Droid today, you're also new to the Android OS from Google. Navigating the sea of apps for the Droid can be a bit daunting. Here's my out-of-the-box suggestions for mandatory free Apps, in no particular order:
1) Facebook: It probably came pre-installed. Mine did. Use it to sync your contacts with. It will grab your friends phone numbers from their profiles, and load you up.
2) Pandora: If you use Pandora already, sign into this app, and your radio stations are at your fingertips. Again, combined with the car dock and charger, you should get a lot of mileage out of this app.
3) NewsRob: Use Google Reader as your RSS reader? NewsRob syncs up to Google Reader, and has all the features that Reader does (Share, Star, etc).
4) Shazam: Can place the name of the song your listening to? Let Shazam identify it for you, as well as tag it in a list for your hunt down on Amazon MP3 later.
5) Google Sky Map: Augmented Reality for the night sky. This is one of those apps that dances on the border of magic and technology. Wow your friends with this one on a clear night.
6) Color Flashlight: Silly little app, but terribly handy. Gives you a bright white screen to use as a flashlight, as well as other colors/effects. Turn on the strobe. Combine with Pandora for portable Raves.
7) ShopSavvy: Another Wow-Factor app. Scan a barcode, check prices on the internet, and a store locations near you via GPS.
8) Google Calendar widget: Use Google Calendar? drop the widget onto one of your three desktops for constant schedule-at-a-glance. If you're syncing with Exchange at work, put the Corporate Calendar widget up right next to it and try and keep your work from bleeding into your play.
9) Twidroid (app) or Twidgit Lite (widget) : Twidroid is generally regarded as the best Twitter client app for the Android, but Twidgit is a widget, not an app, which I much prefer. Take your pick.
10) Google Voice: OK, this one is a big leap for people, but I was a Google Voice user before I got this phone. If you use Google Voice, this phone is heaven. Seamless integration with Google Voice mail, as well as allowing you to dial out from your Google Voice number instead of the real number of your phone. If you're not yet a Google Voice user, now may be the time to jump in. If you want visual voicemail from Verizon, you'll have to from $3/month for the privilege, and it doesn't transcribe voicemails to text like Google Voice does. Otherwise, you'll be connecting to your voicemail box like you always have, spending 45 seconds navigating phone trees to hear your friend leave you a 2 sec message saying "call me".
[Updates and Additions]
Since writing this, I've snagged a few more essentials:
Mother TED: Are you a TED addict? Get your fix directly through an app. It supports tagging, searches, etc. AND IT DOWNLOADS TED VIDEOS RIGHT TO YOUR PHONE! Seriously, what more do you want short of jacking the phone right into your skull?
Weather Channel: A great weather forecast app, with support for widgets. Set a home location and it will appear in the notification bar at the top. So tasty.
Google Listen: Are you a podcast junkie? Look for your favorites on Google Labs' Listen app. No need for syncing your subscriptions with iTunes or Zune anymore, just get them delivered directly to your phone.
WiFiScanner: This is a great app, and has let my Droid replace my netbook and kismet as my WiFi site survey tool. Returns signal strength in dbm, and has an autoscan mode forconstant data updates. Maybe this one is a bit specific to Network Engineers and WarDrivers.
RPN Calculator: Again, over on the geeky side, if you need an RPN stack to use a calculator correctly, this is it. If you don't know what Reverse Polish Notation is, then nevermind, you'll think the app is broken. :-)
gTasks: I'm still on the lookout for a better app to sync my task list from Google (preferably one with widget support). Until then, this clean little app will do the trick.
Did I miss anything? Please share your favorite apps in the comments. Or tell me all the ways that the iPhone is better, and where I can shove my Droid.