Or this:
if(is_home())
{
<div id="contenthome">
<?php } else { ?>
<div id="content">
<? } ?>
Personally, I prefer writing things in shorthand when it comes to cases like these. Consider also using this to shorten your code:
<div id="<?php echo is_home() ? 'contenthome' : 'content'; ?>">
ADDITIONAL NOTES: If your <body> tag has the Wordpress body_class() added to it, you can pretty much target individual pages with CSS, even when they're using the same template. The homepage, like any other page, will have a unique body class applied that will then allow you to target that particular page.
So if your issue is simply a matter of being able to target any given page regardless of template with CSS, you won't necessarily NEED to apply any unique divs to your page. For example:
<style type="text/css">
#content{display:block;}
.home #content{display:none;}
</style>
This will hide the content div only on the homepage. It's probably not what you want it to do, but you can see how the page itself is being targeted to override the default behavior.
UPDATE: As per your latest question, if you need another particular page to use that conditional, use is_page() like so:
LONG CODE:
if(is_home() || is_page('posts_page_title'))
{
<div id="contenthome">
<?php } else { ?>
<div id="content">
<? } ?>
SHORTHAND:
<div id="<?php echo is_home() || is_page('posts_page_title') ? 'contenthome' : 'content'; ?>">
If you prefer, you may also use is_page('posts_page_ID') or is_page('posts_page_slug'). Play around with it and see what works best for you. More information here: http://codex.wordpress.org/Function_Reference/is_page