There are a few different reasons.
Fundamentally, your logical code (i.e. your plugin or your theme) should not need to know anything about the database. At all. Really.
$wpdb object is the global database access layer, and you should be using it for all of your database access. If you need to run a custom query (let's say you have a custom table somewhere) then you should use
$wpdb->query() to prepare your queries and fetch data from the database.
It's very easy to forget to sanitize a query and open your site to some kind of SQL injection attack. Using WordPress' built-in sanitation methods is one way of protecting against this. It's not perfect (you could still write a really bad query) but it definitely helps.
The queries built in to the WordPress API have been poured over by several developers and tweaked to run with the smallest memory footprint and return as quickly as possible. In addition, some of these API calls are self-caching, so you can "query" for data that hasn't changed and is already in memory. This is a huge performance benefit over making a direct SQL call.
And with every new version of WordPress, we make changes to make things faster and more nimble.
4. Flexibility and Stability
The database is not a fixed construct. The table schema can and will change. Hard-coding queries to WordPress objects in your own code is an inflexible maintenance nightmare. When a new version of WP drops, you'll have to rewrite your queries again ... opening yourself up to potential vulnerabilities (#2) again or negatively impacting your site's performance (#3).
The Bottom Line
All of PHP is still available to you. You can make direct connections to whatever database you need and run whatever queries you want. But WordPress has already (in the majority of cases) done the heavy lifting and abstracted these calls for you.
The benefit of using
$wpdb is that it's already there, is maintained by several developers, and is well-documented. You don't have to reinvent the wheel. And you know, if WordPress changes, it will change along with it for you.