Apple's Home app:
Apple's Home app comes pre-installed on iOS, iPadOS, and macOS. Apple's approach is a clean, and simple design, but it sacrifices some advanced features to keep the app simpler for people getting started.
The actual HomeKit framework has more advanced features that can be utilized when using third party applications. Here are some of the more advanced features you can utilize in your HomeKit system when using third party apps.
Room Light Level
Vibration (shown as motion)
Garage door obstruction
Recurring automations (repeat every x minutes or hours)
Revert to previous state after running automation for X minutes
Turn off an automation after running
Undo automation after 12 hours (instead of the 4-hour maximum in the Home app)
Eve is a free iOS app that offers a couple of advantages over Apple's Home app. You can use the Eve app when you want to create more complex automations, but don't want to pay for an app like Home+. Some people may also prefer Eve's interface over Apple's Home interface.
Home+ is a paid iOS app that offers several advantages over Apple's Home app, including: only app to support all of HomeKits automation features, very complex and detailed automations, customizable widgets, a great watch app, more icons, activity log, etc. If you create a lot of automations, the app allows you to name your automations.
Probably the most unique feature of Controller is the ability to backup your HomeKit setup. It also allows what it calls "Control Views" and "Workflows". It also provides logs and other advanced features.
HomePass is a different type of HomeKit app. It is used to manage and store device HomeKit codes. This can be handy for when you have to reset and re-add a device back into HomeKit and can no longer located the device's HomeKit code. It is really well made and has options to scan in codes, export codes, and even has a companion watch app that can be used to display device codes that can then be scanned by your iPhone when adding devices back into HomeKit.
Harmony Bridge is a macOS app which allows you to integrate Logitech's Harmony Remote Hubs into HomeKit. This allows you to add complicated AV controls in HomeKit scenes. An example would be a Harmony macro that powers up your home theater Blu-ray player, then turns on your receiver and changes it's input to the Blu-ray, then powers on your projector and selects Input 1. You can call this macro in a HomeKit scene or automation.