Ah, it turns out I had a couple of things wrong:
1. I'm used to seeing WP_Query terms in "underscore_case" so assumed this is how they'd be stored in the database. I took a look at the wp_term_taxonomy table and it's as it looks on the admin, i.e." hyphen-case". I also had my "tax_query" code a little wrong, as it's an array of an array i.e.
'tax_query' => array( array( 'taxonomy' => 'slider-categories', // the taxonomy group 'field' => 'slug', // To get the taxonomy category, this can also be ID 'terms' => 'homepage-slider', // Your taxonomy category slug 'operator' => 'AND' // This can be IN, NOT IN, or AND ) )
A better explanation of this can be found <a href="http://wordpress.stackexchange.com/questions/50399/only-show-posts-from-a-certain-category">here</a>