First, we need to talk a bit about the history of the simply typed lambda calculus.

There are two historical developments of the simply typed lambda calculus.

When Alonzo Church described the lambda calculus the types were baked in as part of the meaning / operational behavior of the terms.

When Haskell Curry described the lambda calculus the types were annotations put on the terms.

So we have the lambda calculus a la Church and the lambda calculus a la Curry. See https://en.wikipedia.org/wiki/Simply_typed_lambda_calculus#Intrinsic_vs._extrinsic_interpretations for more.

Ironically, the language Haskell, which is named after Curry is based on a lambda calculus a la Church!

What this means is the types aren't simply annotations that rule out bad programs for you. They can "do stuff" too. Such types don't erase without leaving residue.

This shows up in Haskell's notion of type classes, which are really why Haskell is a language a la Church.

In Haskell, when I make a function

sort :: Ord a => [a] -> [a]

We're passing an object or dictionary for Ord a as the first argument.

But you aren't forced to plumb that argument around yourself in the code, it is the job of the compiler to build that up and use it.

instance Ord Char instance Ord Int instance Ord a => Ord [a]

So if you go and use sort on a list of strings, which are themselves lists of chars, then this will build up the dictionary by passing the Ord Char instance through the instance for Ord a => Ord [a] to get Ord [Char], which is the same as Ord String, then you can sort a list of strings.

Calling sort above, is a lot less verbose than manually building a LexicographicComparator<List<Char>> by passing it an IComparator<Char> to its constructor and calling the function with an extra second argument, if I were to compare the complexity of calling such a sort function in Haskell to calling it in C# or Java.

This shows us that programming with types can be significantly less verbose, because mechanisms like implicits and typeclasses can infer a large part of the code for your program during type checking.

On a simpler basis, even the sizes of arguments can depend on types, unless you want to pay fairly massive costs for boxing everything in your language up so that it has a homogeneous representation.

This shows us that programming with types can be significantly more efficient, because it can use dedicated representations, rather than paying for boxed structures everywhere in your code. An int can't just be a machine integer, because it has to somehow look like everything else in the system. If you're willing to give up an order of magnitude or more worth of performance at runtime, then this may not matter to you.

Finally, once we have types "doing stuff" for us, it is often beneficial to consider the refactoring benefits that mere safety provides.

If I refactor the smaller set of code that remains, it'll rewrite all that type-class plumbing for me. It'll figure out the new ways it can rewrite the code to unbox more arguments. I'm not stuck elaborating all of this stuff by hand, I can leave these mundane tasks to the type-checker.

But even when I do change the types, I can move arguments around fairly willy-nilly, comfortable that the compiler will very likely catch my errors. Types give you "free theorems" which are like unit tests for whole classes of such errors.

On the other hand, once I lock down an API in a language like Python I'm deathly afraid of changing it, because it'll silently break at runtime for all my downstream dependencies! This leads to baroque APIs that lean heavily on easily bit-rotted keyword-arguments, and the API of something that evolves over time rarely resembles what you'd build out of the box if you had it to do over again. Consequently, even the mere safety concern has long-term impact in API design once you ever want people to build on top of your work, rather than simply replace it when it gets too unwieldy.