Having used the BlackBerry Z10 for over a week now, I can now compare its built-in apps and third-party apps to the iPhone 5's. The iPhone 5 and the BlackBerry Z10 both have solid and visually appealing built-in applications. The BlackBerry Z10 seems to have the edge in the overall look and quality of the apps built-in, and this is mostly to due to Apple not really updating or making any major changes to the built-in apps it ships with the iPhone 5.
BlackBerry is known for its keyboards and also its email client, and both are excellent on the Z10. The way that the BlackBerry Z10 handles email and more is through its BlackBerry Hub. BlackBerry Hub is similar to the drop-down notification center built into iOS. It is the only place you need to go to access all of your emails, messages, notifications, social media updates, appointments and more. I find this quicker to use than opening Apple's separate email app, messaging app, Facebook app and Twitter app. With the BlackBerry Z10's Hub feature you can get access to features that you would need to open several separate apps for on an iOS device.
When it comes to Web browsing, BlackBerry and Apple's mobile Web browsers are both based on the WebKit engine. I found that the BlackBerry Z10 loaded pages just a little bit faster than the iPhone 5. Web browsing on both smartphones offers quick Web page renderings, but the Z10 just edged out the iPhone 5.
One of the coolest features of BlackBerry 10 is how it handles multitasking and how it maintains displaying all app information, compared to iOS. For instance, if you open the weather app on the Z10 you get the forecast, but when you minimize the app it pins it to a page and it constantly updates itself so you can always see what the current weather is; it is very similar to Windows Phone 8's Live Tiles. When you open the weather app on an iPhone 5 and close it, you'll need to reopen it from the app or Notification Center. Hopefully Apple will add similar features to iOS 7.
When it comes to third party apps, there is no competition. The iPhone 5 and Apple's App Store put the BlackBerry Z10 and BlackBerry World to shame. There aren't many popular third-party apps available for the BB10 platform yet. This number will definitely grow over time, but for now it's slim pickings in terms of apps and quality of apps.
As far as which smartphone opens apps faster, that would go to the iPhone 5. Apps open almost instantaneously on the iPhone 5 compared to the BlackBerry Z10. A possible reason for this is that BlackBerry has added a ton of animations to BlackBerry 10 to make it visually appealing, but that seems to slow things down in certain areas. I'm sure BlackBerry will release an update in the future to optimize the device to make it faster in opening apps.
The iPhone 5 and the BlackBerry Z10 are both excellent smartphones and I wouldn't have a problem recommending either. I just think it might be best to let BlackBerry 10 mature a little bit, get more quality third-party apps, and then it would be a worthy competitor to the iPhone 5.