So I think that the solution to what you're asking is found in the filter priority that both WP Biographia and AddThis use to add additional content after the post content.
Both plugins use
add_filter to hook into the WordPress
the_content filter. WP Biographia does this in
/wp-plugin-base/wp-plugin-base.php at line 27, called from
wp-biographia.php at line 136. AddThis does this in
addthis_social_widget.php at line 90.
By default, both plugins hook
the_content at the default priority of
10; this means that the order of the plugin's content being appended your post content is done on a first come, first served basis, dependent on which plugin effectively got there first (usually this is alphabetical but there are other factors which can affect this).
AddThis doesn't allow you to tweak the
the_content filter priority but WP Biographia does. This is documented in the Advanced Settings section of Bending WP Biographia To Your Will, though you may not have realised it; the post says this ...
Finally, the plugin uses the standard WordPress
the_excerpt filters to add the Biography Box to your posts and pages, either at the start or at the end of the content. As this is a standard WordPress feature, you may find other plugins and/or themes do exactly the same thing, which means that if you have multiple additions to a post or a page via this mechanism you may not end up with the additional content in the order you want. The Content And Excerpt Priority Settings allows you to adjust the filter priority to get the order you really want; a lower priority means the plugin’s filters will fire earlier, thus bumping them up the order in which the additional content is added, a higher priority will fire later, with the opposite effect.
From the behaviour you've described, it sounds like changing the
the_content filter priority (and possibly the
the_excerpt filter priority as well, depending on your site's theme) to a value higher than the default value of
10 will do what you want. You may need to play around with the precise value of the filter priority, but I'd start with priority
11 and see if that has the desired effect. This should (fingers crossed) make WP Biographia append the Biography Box later than AddThis, effectively making the Biography Box appear after the output of AddThis.
Let me know how you get on.