I don’t now. The relevant stuff in the case here is a better/different Unicode handling by bash 4, told me the author of Translate Shell.
Besides that I have no clue what the major differences are. I also worked with bash 3 till now.
Here are the complete release notes covering bash 2 through 4.4. — I’m going to read them… one of these days 
If so, does it completely replace the current version?
You can uninstall it at any time with brew uninstall bash. Then the old, preinstalled bash 3 will be used. (Bash 4 via Hombrew will go to /usr/local/bin, whereas the system’s bash 3 is in /bin. So nothing gets overwritten or so.)
Is there anything special I need to do after the upgrade?
Hombrew says, for full usage in the interactive shell the path of bash 4 should be added to /etc/shells. I think this is a security thingy.
As long as you have set your Terminal to use the default login shell (see screenshot below) and /usr/local/bin is on your PATH it will be used by the Terminal automatically.
Terminal prefs, General tab:
Similar of course for KM: /usr/local/bin must be in your ENV_PATH variable. But I guess it is there already, otherwise KM would not find any of your Homebrew-installed tools.