If I have a function parameter WITHOUT the type constraint:
> function a ($s=$null) {if ($s -eq $null) {Write-Host "HI"} if ($s -eq "") {Write-Host "KK"}}
> a
HI
Now if I add the type constraint to it, the $null is interpreted differently:
> function a ([string]$s=$null) {if ($s -eq $null) {Write-Host "HI"} if ($s -eq "") {Write-Host "KK"}}
> a
KK
I can't find doc that explain this. It's also not consistent.
In your first example (function a), $s is equivalent to $null - it's truly null.
In your second example (function b), because you're casting $s to a [string] object, it's actually an empty String (equivalent to [String]::Empty), not $null.
You can check this by adding the following to each of your functions:
if($s -eq [String]::Empty){"empty!"};
Only b will print empty! - a will evaluate this to $false
Alternately, add this:
$s|get-member
a will actually throw an error - the same error you'll get if you run $null|get-member. b will show you that $s is a string and list all of the members of that class.
nullto a string will result in string.Empty in C#. But then there's no requirement that PowerShell be consistent with C# as they're different languages. Similarly, F# also has different behaviors than C#, and it's by design.