We've known for a while that your smartphone is a better PC than your PC ever was. It can't replace everything your computer does, but there are some things that are actually better to do on your phone rather than on your computer.
It should be noted that much of this is going to be subjective. There huge differences just between iOS, Android and Windows Phone, much less individual handsets. However, the prevailing notion has always been that phones are great when you're away from the computer, but if you have a PC handy, you should use that. For things like photo editing or watching movies, that might be true. Here are some things that, more often than not, can be done more productively on your phone than your computer.
Reading (but Not Answering) Email
Email is a chore, no matter who you are. However, most of us probably receive more email than we reply to. Plenty of it is either junk, spam or just needless notification. Your phone is a way better place to get rid of it as it comes in than your computer is. On Android, for example, you can archive an email directly from the notification it came from. If you know you don't need to read it, away it goes.
Replying to email is considerably harder (mostly due to the crappy, tiny keyboards), but the upside is that if you get rid of the junk as you see it, you have much less to reply to when you get back. It's also totally optional. You don't need to constantly be checking your phone for new email but, if we're honest, you're probably looking at it every few minutes anyway. You may as well get rid of the junk.
Checking Your Bank Balance and Depositing Cheques
Most mobile apps for banks have had to adapt to the limited input functionality with special security mechanisms. You can have your phone remember your password on most services, but a bank needs to authenticate you every time. So, they require a PIN every time you open the app. The result, however, is that it's stupid simple to check your bank balance on your phone.
Additionally, many banks now allow you to deposit cheques via your smartphone app. While some computers have webcams installed, they're not really for taking pictures of things. Point goes to the smartphone, which can take a picture of both sides of your cheque, deposit your money, and be done before you can recall your mother's maiden name.
Of course, this depends somewhat heavily on your bank's mobile app (not all banks invest equally into mobile technology), but many are starting to finally enter the digital age. At the very least, it's worth checking out what online services your bank offers when choosing a bank.
Setting and Receiving Reminders
Software to remind yourself to do things later has existed as long as computers have, but it's never been quite this good. Because your phone is always with you, there's little chance that you're going to miss a notification. No matter what app you use, the very nature of having your phone nearby makes it hard to miss.
It gets even better when you factor in voice reminders. Apps like Google Now allow you to set reminders in natural language. Not only can you can trigger them to remind you about something at a certain time, you can also get reminders when you arrive at a particular location (like home or work). Even the most sophisticated PC software can't manage that while it's sitting in your laptop bag, or plugged into your desk.
Quick Google Searches for Simple Facts
Arguably, this may be more of a function of voice commands than smartphones per se, but if you want to find out how tall Jeff Goldblum is, the computer is probably clunkier. With some clever tweaking, you can get voice commands working smoothly on the desktop, but most people don't. Both iOS and Android, however, allow you to speak out a sentence and get useful facts back.
Saying "how many ounces are in a quart" is inarguably faster than typing it and, let's be real, most people aren't talking to their desktops. You can do lots of stuff with Google's voice commands and Siri is just as helpful.
Reading Through Your News Feeds
There's a reason you don't take your laptop into the bathroom with you anymore. Your smartphone is clearly better (most of the time) for reading through feeds. Not only because scrolling with a touch screen is smooth and natural, but they're much lighter and easier to carry with you.
In fact, reading content on your mobile device is so nice, they built an entire product segment around making it even better: tablets. Functionally, there's almost no difference between a smartphone and a tablet. Except the bigger screen size. While most of us don't want to carry a 7-10-inch phone around in our pocket, we love the idea for sitting on the couch in the living room reading. Laptops are ok if you want to multitask with something productive too, but if you're just leaning back and reading, nothing beats a phone (or tablet).