Can someone tell me if either of these two methods has an advantage over the other and why?
$mydir = ABSPATH.\'/wp-content/themes/mytheme/images\';
ABSPATH) is not recommended. See the Determining Plugin and Content Directories WordPress Codex article.Since PHP 4.0.2, symlinks are being resolved for the __FILE__ and __DIR__ magic constants, so take that into account.
Bottom line: To determine the absolute path to a theme directory, I would suggest to use the get_template_directory() function which also applies filters and internally combines get_theme_root() and get_template().