This question leans towards getting some subjective answers, as I think it’s a matter of personal preference. The built-in editor works just as well as any other text editor. That goes for any of your theme files, not just header.php.
The benefit of using an external editor (like Notepad++ and using FileZilla to FTP your changes), is that you have the possibility of using ‘syntax coloring’.
I highly recommend Notepad++, as it supports multiple document encodings (UTF-8, ANSI, etc) and many scripting languages (with their generally accepted syntax coloring), such as PHP, JavaScript and CSS, your most common languages when working with WordPress.