Is NULL in C required/defined to be zero?

NULL is guaranteed to be zero, perhaps casted to (void *)¹.

C99, §6.3.2.3, ¶3

An integer constant expression with the value 0, or such an expression cast to type void *, is called a null pointer constant.(55) If a null pointer constant is converted to a pointer type, the resulting pointer, called a null pointer, is guaranteed to compare unequal to a pointer to any object or function.

And note 55 says:

55) The macro NULL is defined in <stddef.h> (and other headers) as a null pointer constant.

Notice that, because of how the rules for null pointers are formulated, the value you use to assign/compare null pointers is guaranteed to be zero, but the bit pattern actually stored inside the pointer can be any other thing (but AFAIK only few very esoteric platforms exploited this fact, and this should not be a problem anyway since to "see" the underlying bit pattern you should go into UB-land anyway).

So, as far as the standard is concerned, the two forms are equivalent (!ptr is equivalent to ptr==0 due to §6.5.3.3 ¶5, and ptr==0 is equivalent to ptr==NULL); if(!ptr) is also quite idiomatic.

That being said, I usually write explicitly if(ptr==NULL) instead of if(!ptr) to make it extra clear that I'm checking a pointer for nullity instead of some boolean value.

1. Notice that in C++ the void * cast cannot be present due to the stricter implicit casting rules that would make the usage of such NULL cumbersome (you would have to explicitly convert it to the compared pointer's type every time).

From the language standard:

6.3.2.3 Pointers
...
3 An integer constant expression with the value 0, or such an expression cast to type void *, is called a null pointer constant.⁵⁵⁾ If a null pointer constant is converted to a pointer type, the resulting pointer, called a null pointer, is guaranteed to compare unequal to a pointer to any object or function.
...
55) The macro NULL is defined in <stddef.h> (and other headers) as a null pointer constant; see 7.17.

Given that language, the macro NULL should evaluate to a zero-valued expression (either an undecorated literal 0, an expression like (void *) 0, or another macro or expression that ultimately evaluates to 0). The expressions ptr == NULL and !ptr should be equivalent. The second form tends to be more idiomatic C code.

Note that the null pointer value doesn't have to be 0. The underlying implementation may use any value it wants to represent a null pointer. As far as your source code is concerned, however, a zero-valued pointer expression represents a null pointer.

In practice is the same, but NULL is different to zero. Since zero means there's a value and NULL means there isn't any. So, theoretically they are different, NULL having a different meaning and in some cases that difference should be of some use.