I am pretty unsure if my explanation and example really cover the most important principles
Think of behavior as scenarios external to the structures. A certain data structure can "be used" in multiple behaviors/scenarios.
On the other hand think of structure related logic as being internal to the structure. The structure gets affected by various changes and executes some actions as a consequence.
That being said we can exemplify with the following:
Structure design patterns will define a weblog by defining its constituents as higher level business objects such as Article/Image/Comment. The constituents are aware of one another and how to connect to each other.
$image = new Image;
$image->title = 'Image title';
$image->url = 'http://example.com/file.ext';
$image->save(); // will save the image to a DB
$article->title = 'The title i have set';
/* $article->url_key = 'the_title_i_have_set'; */
// this ^^ element of logic will be done automatically by the article
$article->addImage($image); // will save the connection between the
// article and the image to DB
Behavior design patterns will define a weblog by its use cases (scenarios) using lower level business objects such as Article/ArticleToImage/Image/ArticleToComment. The business objects are not aware of each other and are "maneuvered" into place by the scenario logic.
$image = new Image;
$image->title = 'Image title';
$image->url = 'http://example.com/file.ext';
$image->save(); // will save the image to a DB
$article->title = 'The title i have set';
$article->url_key = $controller->getURlKey($article->title);
$article->save(); // saves article to DB
$article_to_image = new ArticleToImage;
$article_to_image->article = $article;
$article_to_image->image = $image;
$article_to_image->save();
TL;DR
If the storage objects are smart (contain logic) that's structural design. If the storage objects are dumb (they can only store data and transfer it to the DB) you then need a behavioral design to manage them.