Let's collect knowledge like we did with previous guides - paste in your notes and share what you've learned, and somebody else will organize it later. :) If you want to ask questions and share tips over chat with other developers, see How to use IRC for how to connect to #theos and #iphonedev.

Unlike previous instances of "Updating extensions for iOS X", this specific version has several internal changes as well as a different public jailbreak method which demands a separate page for it.

What has changed in iOS 9.3.3? (Classes, frameworks, etc.)

SBUIController's -finishLaunching ceased to exist. Consider using notifications to perform initializations of your tweaks if you were hooking this method. (com.apple.accessibility.system.app.server.ready, com.apple.frontboard.systemappservices.serverNotifyToken, SBSpringBoardDidLaunchNotification, com.apple.springboard.finishedstartup, com.apple.sharing.SpringBoard.started, com.apple.springboard.homescreenunlocked are some of the notifications to register for.)
SpringBoard's -_relaunchSpringBoardNow ceased to exist. Use [[FBSystemService sharedInstance] exitAndRelaunch:YES] (AFAIK there is no difference between YES and NO at the end).
To reboot, use [[FBSystemService sharedInstance] shutdownAndReboot:YES] and to shutdown, use [[FBSystemService sharedInstance] shutdownAndReboot:NO]
SBOrientationLockManager's -isLocked ceased to exist. There is now -isUserLocked (without lock override check) and -isEffectivelyLocked (with lock override check).
SBScreenShotter has been replaced with SBScreenshotManager. An instance of SBScreenshotManager is available through SpringBoard's screenshotManager property. To take a screenshot, use SBScreenshotManager's saveScreenshotsWithCompletion: method
SBWallpaperController now applies styles based on the current bundleIdentifier, rather than globally. For example: [wallpaperController setHomescreenStyle:1 priority:5 withAnimationFactory:nil]; becomes [wallpaperController setHomescreenStyle:1 forBundleIdentifier:@"com.apple.SpringBoard" withPriority:5 animationFactory:nil];