I am on your side as a customer. If you prefer to use the web site, then the company should support this decision.
That being said, as a software developer, there are lots of advantages to having people use the app. As Jespah says, when I develop an app I have control over the screen and can provide a much cleaner, faster interface.
From a marketing perspective, companies want to be on your mobile device. That way you will always have Target in your hands.
I have a Starbucks logo, a bank logo and an Amazon logo staring at me any time I open my cellphone. However, I find each of these apps to be useful, so I keep them around.