Here are some of the apps I use the most....
Apex Launcher - The first thing I do when I get a phone or tablet is install Apex. Complete control of over the number and size of icons that can be put on the screen. Many gestures available. Just a wonderful interactive experience that is consistent from phone to tablet.
Pebble - Yes, I have a Pebble watch and love it.
Drippler - This app is a news aggregate app that has news stories specific to the device on which it's installed.
Team Viewer - A remote control app to let me work on my PC from anywhere.
Groove IP - This app works in conjunction with Google Voice that lets me make/receive phone calls on my wifi-only Note. When a person calls my Google Voice number it rings on my phone, PC, tablet, home phone and office phone. Ironically, the tablet and PC start ringing about 3 seconds before the cell phone rings.
Google Voice - This is my main text messaging app.
Touchdown - Connects to my company's corporate Exchange email server.
Samsung Milk - Samsung's new music app that I really like. It's not in the Play Store for the Note so I had to search XDA forums for an APK download.
Chrome Beta browser - I just like it better since it easily syncs bookmarks with my desktop.
Timely - This is a clock replacement that I use for my alarm clock.
Auralux - This is a fascinating game that's very hard to explain. If you have Osmos then you kinda know what Auralux is like. Try it.
Pulse - Another news aggregater.
Droid of the Day - A daily app finder
Amazon app store - A paid app for free every day.
Fancy Widgets - Nobody has more weather widget skins than them. Most of them are very pretty.
PushBullet - It's the easiest way to pass links, text, etc. from device to device to PC. If you've been emailing links to yourself to get something from your PC to your tablet then you need PushBullet. It also has the side benefit of being able to display your Android notifications as fly-outs on your PC's Chrome browser. You won't need to pick up your phone/tablet to see what your phone/tablet is buzzing about.
Slice - I buy a lot of stuff online. Slice will parse through my email and keep me up to date on where that stuff is at in the supply chain. And, since it knows everything I bought I don't have to search my email to find a receipt if I want to buy the same thing again.
Wakelock Detector - If your battery is draining then this app will pinpoint which apps are waking up the CPU. Very handy.
Droid Over Wifi - Want to send/receive files to/from your tablet without plugging it into your PC? This is for you.
PicSpeed Wallpapers - This app has the most beautiful wallpapers around. They add dozens daily.
Wallpaper Changer - Combine this with PicSpeed and your background will never be boring.