Auto Layout vs Frame Sizes
Relying on frame sizes equals calculating the sizes. When using percentage or proportions, it could work on different phone models, but still might look sometimes strange on few resolutions. Then you would add another conditional statement because e.g. on iPhone 4s you would want a little bit different layout. Your code is more and more complicated and could lead to the place where any change is a risk. It is doable, but why reinventing the wheel?
Tools like basic constraints replace sometimes really huge logic blocks for layouts. Logic to compress one label when second one is extending means few ifs. Think about 10 labels. With addition of size classes, enabling/disabling constraints, compression & hugging, auto layout has become irreplaceable for me and I would suggest to take a look at it since most of companies are using it, if not in storyboards, then in xibs or in code, but with auto layout.
tl;dr Learn how to use Auto Layout – it's a huge time saver for real-world apps!
Long answer:
Finally it is up to you if you want to use features like Auto Layout or Size Classes that Apple provides. The main benefit of those is not really performance in terms of UI rendering in the final product. It is the time you need to invest into thinking about all edge cases if you plan to develop an app that works on different screen sizes and orientations.
Let's say you want to do all the work you need to do to support iPhone 4/4s screen size, iPhone 5/5s screen size, iPhone 6/6s (Plus) screen size and iPad screen size plus Portrait and Landscape modes for all the above screen sizes yourself (here's a good overview): Well, there's nothing wrong with it, just go for it. You would then, of course, need to test things very carefully and also always keep this list up to date – in each app you are using your own solution. If that's for you, do it.
But most people want to get their ideas out without struggling with things like different screen sizes too much. You could of course extract your own logic from one app to use it in all of your apps, and that's exactly what Auto Layout and Screen Sizes are about. Simply put, Apple has done all of the work to make your frame-setting work in different screens for you. You just need to use their new vocabulary (Constraints) to make it work.
In other words: It's an abstraction layer on top of handling with the screen rendering logic directly. If you don't like it, let it go. But if you're planning to do some serious apps that should work on different iPhone/iPad generations and also plan to maintain those however, then, please, learn how to do things with Auto Layout and Size Classes. It will save you and all future maintainers quite some time in development.
A good starting point is the docs. Or a tutorial like this one on raywenderlich.com.
Apple says the following about this difficulty themselves (in the docs linked above):
In many ways, programmatically defining a view’s frame provides the most flexibility and power. When a change occurs, you can literally make any change you want. Yet because you must also manage all the changes yourself, laying out a simple user interface requires a considerable amount of effort to design, debug, and maintain. Creating a truly adaptive user interface increases the difficulty by an order of magnitude.
BTW: There's no difference at all in using Auto Layouts or Frames regarding the programming language: Both Swift and Objective-C support both ways perfectly well. It seems you just didn't find out how to change frames in Obj-C yet. ;)
One more thing: You also don't need to use Storyboards to use Auto Layout. You can use Auto Layout from code. See the docs here. There's even plenty of frameworks trying to make this easier (the original APIs from Apple tend to be not very pretty in many terms), I can recommend SnapKit.