You currently have to load all third-party apps from a paired iPhone over Bluetooth, which isn't ideal when you don't have your iPhone around. It also creates longer loading times for opening and using apps, which should be alleviated considerably once Apple Watch apps are stored directly on the device.
Apps running natively will also allow them to do things like access the Watch's heart rate sensor, which will make it easier to use a workout app like Strava.
The Apple Watch will also be able to connect to your saved WiFi networks independently from a paired iPhone, which means your iPhone can be turned off in the house and your Watch can still be connected to the internet.