There are three main areas of difference.
First, password management is a feature for browser developers, while it is the entire product for the third parties. So typically, the browser managers only offer basic core features. The 3rd parties add lots of useful stuff on top to differentiate their product. For example:
- Storing other information apart from web page logins.
- More sophisticated password generation features.
- Checking sites against lists of breaches and warning you to change your password
Second, because 3rd party managers are not tied to the browser, there are some situations in which they are more flexible. for example:
- You can have one password store that works with multiple browsers on your machine
- The browser managers typically only use the browser's services for backup and sync. For example, Chrome only uses Google Sync Services, Safari only uses iCloud. If you want to use Dropbox, you have to go third party.
- The portable/mobile versions of 3rd party managers will show you the password if required, so you can type it in on a temporary machine without having to configure the browser to sync all your passwords over and then remove them later.
Lastly, third party managers are usually targeted at the security-conscious and so usually better document their security practices and have security as higher on their list of priorities.