The preferred way of creating a new element with jQuery

I personally think that it's more important for the code to be readable and editable than performant. Whichever one you find easier to look at and it should be the one you choose for above factors. You can write it as:

$('#box').append(
  $('<div/>')
    .attr("id", "newDiv1")
    .addClass("newDiv purple bloated")
    .append("<span/>")
      .text("hello world")
);

And your first Method as:

// create an element with an object literal, defining properties
var $e = $("<div>", {id: "newDiv1", name: 'test', class: "aClass"});
$e.click(function(){ /* ... */ });
// add the element to the body
$("#box").append($e);

But as far as readability goes; the jQuery approach is my favorite. Follow this Helpful jQuery Tricks, Notes, and Best Practices

Much more expressive way,

jQuery('<div/>', {
    "id": 'foo',
    "name": 'mainDiv',
    "class": 'wrapper',
    "click": function() {
      jQuery(this).toggleClass("test");
    }}).appendTo('selector');

Reference: Docs

Be sure to read the docs thoroughly, as this notation has certain consequences that won't be immediately obvious to the person inspecting the code.

The name "class" must be quoted in the object since it is a JavaScript reserved word, and "className" cannot be used since it refers to the DOM property, not the attribute. While the second argument is convenient, its flexibility can lead to unintended consequences (e.g. $( "<input>", {size: "4"} ) calling the .size() method instead of setting the size attribute).

According to the jQuery official documentation

To create a HTML element, $("<div/>") or $("<div></div>") is preferred.

Then you can use either appendTo, append, before, after and etc,. to insert the new element to the DOM.

PS: jQuery Version 1.11.x

According to the documentation for 3.4, It is preferred to use attributes with attr() method.

$('<div></div>').attr(
  {
    id: 'some dynanmic|static id',
    "class": 'some dynanmic|static class'
  }
).click(function() {
  $( "span", this ).addClass( "bar" ); // example from the docs
});

It is also possible to create a div element in the following way:

var my_div = document.createElement('div');

add class

my_div.classList.add('col-10');

also can perform append() and appendChild()

I would recommend the first option, where you actually build elements using jQuery. the second approach simply sets the innerHTML property of the element to a string, which happens to be HTML, and is more error prone and less flexible.

If #box is empty, nothing, but if it's not these do very different things. The former will add a div as the last child node of #box. The latter completely replaces the contents of #box with a single empty div, text and all.