New apps

Reminders

New system-wide features

Notification Center and unobtrusive notifications
iCloud
Split keyboard (iPad only)
LED Flash on incoming calls and alerts
Custom Vibration Patterns
Multitasking Gestures (iPad only)
New Twitter Keyboard with # and @ next to Space
iPod app split into Music and Videos (iPhone only)
Emoji emoticons installed by default
Wifi syncing
Use the device while syncing
Over-the-air updates
Set custom alerts for email, calendar, voicemail, and more
See how much space each app is taking up
Delete each app's data
iBooks-style dictionary definitions across iOS
AirPlay mirroring (iPad only)
Custom keyboard shortcuts (autoreplace)
Rounded On/Off toggles

New features in old apps

Mail
Rich-text formatting
Safari
Private Browsing
App Store
Maps
Alternate Routes
Print
Calendar
Week view in landscape mode (iPhone and iPod Touch)
Camera
Accessible from lock screen by double-tapping home
Use volume-up button as shutter
Photo editing
Game Center
iBooks
Settings
FaceTime
Mid-call invitation alerts
FaceTime button in iMessage messages at the top now
You can set the caller ID for FaceTime calls, so if you call someone, you can choose it to display your cellular number or email address.
Improved Facetime Call quality
Photos
Create and manage new photo albums on the device
Videos (iPad-only)
Photo Booth (iPad-only)
iTunes
Notes
Calculator (iPhone-only)
Phone (iPhone-only)
Delete individual recent calls
Messages
iMessages
FaceTime button in iMessage messages at the top now
Send button is now green and reciptient’s text bubble has a slightly lighter color
Contacts
Weather
Hourly, 'My Location' (via GPS) forecast