What's the difference between Array(1) and new Array(1) in JavaScript?

With Array, both are equivalent. The new is injected when it's called as a function:

15.4.1 The Array Constructor Called as a Function

When Array is called as a function rather than as a constructor, it creates and initialises a new Array object. Thus the function call Array(…) is equivalent to the object creation expression new Array(…) with the same arguments.

From ECMA-262, 3th Edition (with similar in 5th Edition). See also 22.1.1 The Array Constructor in ECMA-262 ECMAScript 2020 specification (11th Edition).

According to Javascript: The Definitive Guide (5th Edition), page 602, "When the Array() constructor is called as a function, without the new operator, it behaves exactly as it does when called with the new operator."

The difference lies in the implementation of the Array function. Whether a call to Array without a new operator will return an instance of Array or not is implementation dependent. For example Mozilla's SpiderMonkey engine does this:

static JSBool
Array(JSContext *cx, JSObject *obj, uintN argc, jsval *argv, jsval *rval)
{
     jsuint length;
     jsval *vector;

     /* If called without new, replace obj with a new Array object. */

That is an actual comment from the actual source. Next lines of code are not reproduced here. I would suppose other engines do the same. Otherwise the behavior is undefined. A good read on this topic is John Resig's post here.

new creates a new object (class instance) which is passed to constructor function as this. But some functions detect if they were not called using new and behave the same way as if they were. Array is one of them, so there is no any difference.

If you want to create such constructor yourself you can do it like in following code:

function Smth(val) {
  if (!(this instanceof Smth)) {
    return new Smth(val);
  }

  this.val = val;
}