Both these code snippets include a conditional tag which removes the site description from the front page.
You can modifiy, replace or add to the conditional tag based on your own needs.
Alternatively, you could also use code like this to return something else conditionally. In this example it simply returns the description.
If you prefer using CSS code, you can grab the post/page i.d or template class from the body classes in your source code and use it in a new rule like this:
.home .site-description {
display: none;
}