What's in an app? Sometimes, I feel that having my favourite program as an app on my phone or tablet is an incentive to use the program itself.
Consider this: If you have the app on your phone, and it also syncs with the program in your computer and the program's web site, it could be really convenient.
The app could also be a third-party program that is authorised to access your data. The simplest example is the email app. If the email app on your phone can sync your emails, you feel confident that you can access your mails anywhere. More so if the app is from the email provider itself - like Gmail. However, I have stopped using email providers that require a payment if I need imap or pop access. Some of them allow access through their mobile site, but you have to go through a browser and it's not convenient. .
It is the same for productivity apps. I like One Note, a note taking app from Microsoft. I have it on my PC, but I can't access it on my Android smartphone. But if there is an app that syncs with my One Note, I would definitely appreciate it. Unfortunately, because there isn't, I use Evernote, which has a desktop program, web-based client and a mobile app. Evernote is excellent for productivity and it also syncs across all the platforms.
Microsoft has a mobile app for One Note, but it is only for Windows 7 phones. It recently released an app for iPhone. I may switch to One Note again if it comes out with an Android app.
Even for social media such as Twitter and Facebook, it's not half as fun if there is no app and you have to start up your browser every time you want to access them. Twitter owes much of its success to mobile and tablet apps like TweetDeck for instance.
Mobile apps are so convenient that I've even switched my primary bank, because it has a mobile app through which I can do most of my transactions.