You might even try Textra. A great, highly recommended SMS app that is light and clean and works really well. Handles MMS and group messaging better than many alternative apps. And I've tried many of them - GO SMS, Handcent, etc. Also the devs are extremely engaged in the app and rarely does more than a week go by w/o an update for the app. You can set not only different tones for the sms notification, you can even set a different color or notification icon (i.e. I have it show an orange icon when someone texts me, but when my wife texts, it's purple). Easy way to see if it was a text from her w/o unlocking the device.
By the way, if you are on 4.3 already, I found a nifty app called NILS. It's a lock-screen widget that you can set up under Settings, Personalize, Lock Screen Style, choose the last one (Widget). Then Choose Settings and choose the widget for NILS. This allows a notification to appear on your lockscreen for any app that sends you a notification. It will remain there until the notification is cleared from the notification panel. You can tailor it for how ever many or few apps you wish to display on your lock screen. So for example, I have Textra, Med Helper, calendar and a select few others to show on my lockscreen. But not my email, Facebook, Wifi notifications, etc. Only my most urgent ones pop up there. When you have notifications, you can swipe down on them to see more details. You can even go directly into NILS settings from there to modify the notification for that specific app. In the settings, you can modify it to hide the clock, day, etc so that it's not duplicating what HTC retains on the widget lockscreen. When you get notifications, it will be somewhat similar in nature to what you were used to with the iphone.
The only drawback on using it on the HTC One - for some reason HTC has locked down the lockscreen. So if you want to open the notification, you can swipe down, choose that app notification - but then you still have to swipe up on the HTC Unlock icon before the app opens. Not sure why HTC makes you do that and doesn't allow a direct link. But I've noticed that seems to apply to any widget that I've tried on the lockscreen.