Specific css on homepage, different one for other pages

Yes there is. You can add this to your themes header.

<?php
  if(is_home()){
    // we are on the home page
    echo '<link rel="stylesheet" src="https://wordpress.stackexchange.com/questions/17113/your_stylesheet" />';
  }
?>

You can also use other conditional tags to find out if you are on nearly every type of WordPress page. Conditional Tags

Following on from the comments below, this would be better used like this

<?php
  if(is_home()){
    // we are on the home page
    echo '<link rel="stylesheet" src="https://wordpress.stackexchange.com/questions/17113/your_home_stylesheet" />';
  }else {
    echo '<link rel="stylesheet" src="your_default_stylesheet" />';
  }
?>