What is the difference between "manage_{$post_type}_posts_columns" and "manage_edit-{$post_type}_columns"?

They both essentially do the same thing.

manage_edit-{$post_type}_columns used to be the old recommended hook used to create or change columns for custom posts types.

With WordPress 3.1 (23 February 2011), manage_{$post_type}_posts_columns was added and is now the recommended hook to use.

Here is a quote from the Codex:

In WP 3.1, manage_edit-{$post_type}_columns has been supplanted by manage_{$post_type}_posts_columns.

supplant = supersede and replace.

Although the old hook is not deprecated, the new hook is the one to use now.

It has to do with the way the admin post table is constructed. WordPress uses a class WP_Posts_List_Table to generate and handle that table. You'll notice that there are a few tables in the admin area similar: Plugins, Terms, Users, etc. They each are their own class, but they all extend from a base class of WP_List_Table.

manage_edit-{$post_type}_columns is a more generic filter that is set (but not called) in WP_List_Table. And, it is actually manage_{$screen->id}_columns. The callback is the get_columns method for the table object.

add_filter( "manage_{$this->screen->id}_columns", array( &$this, 'get_columns' ), 0 );

This filter is used by the get_column_headers() function for retrieving the columns.

So where does manage_{$post_type}_posts_columns then? Remember the method get_columns() that is call for the filter manage_{$screen->id}_columns, the WP_Posts_List_Table class over-rides the base method from WP_List_Table. In WP_Posts_List_Table::get_columns() is where the filter manage_{$post_type}_posts_columns is called.

To recap:

1. A WP_Posts_List_Table object is created
2. It's parent class WP_List_Table adds $this->get_columns() to the hook manage_{$screen->id}_columns
3. The function get_column_headers() calls the filter manage_{$screen->id}_columns
4. WP_Posts_List_Table has over-ridden the method get_columns()
5. Inside of WP_Posts_List_Table::get_columns(), it calls the filter manage_{$post_type}_posts_columns This is where your manage_{$post_type}_posts_columns hook would run
6. WP_Posts_List_Table::get_columns() will return the columns and then run the remaining filters for manage_{$screen->id}_columns

Which one should you use? I don't know. We know that manage_{$post_type}_posts_columns can only be called for post types, whereas manage_edit-{$post_type}_columns may match a different admin page (although I'm not sure if it would).