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.