CSS: Replacing a text on hover, but smooth transition to the new text does not work?

Here is a really simple sample

(This hidden snippet use pseudo elements instead of the visible's inner div's)

div, div:after, div:before {
    height: 100px;
    width: 100px;
    position: absolute;
    font-size: 40px;
    color: black
}

div:after {
    content: "New";
    opacity: 0;
}
div:before {
    content: "Old";
    opacity: 1;
}

div:after, div:before {
    transition: opacity 0.5s linear;
}

.wrap:hover:before {
    opacity: 0;
}
.wrap:hover:after {
    opacity: 1;
}
<div class="wrap">
</div>

div {
    height: 100px;
    width: 100px;
    position: absolute;
    font-size: 40px;
    color: black
}

.new {
    opacity: 0;
}

.old, .new {
    transition: opacity 0.5s linear;
}

.wrap:hover .old {
    opacity: 0;
}
.wrap:hover .new {
    opacity: 1;
}
<div class="wrap">
<div class="new">New</div>
<div class="old">Old</div>
</div>


OK, so first part, you cannot animate the display property, you need a work-around. To do this we fall back to what we can animate, opacity and width/height

For what you are trying to accomplish, I'd use two spans inside the <h1> - one with each text version. Since spans are inline elements we give them display: block so we can control there dimensions more cleanly.

.my_div {
  background-color: red;
  transition: all 500ms ease-in-out;
}
.my_div:hover {
  background-color: green;
}
h1 {
  overflow: hidden;
}
.old-text,
.new-text {
  display: block;
  overflow: hidden;
  transition: all 500ms ease-in-out;
}
.old-text {
  height: auto;
  opacity: 1;
  width: auto;
}
.new-text {
  color: #fff;
  height: 0;
  opacity: 0;
  width: 0;
}
.my_div:hover .old-text {
  height: 0px;
  opacity: 0;
  width: 0px;
}
.my_div:hover .new-text {
  height: auto;
  opacity: 1;
  width: auto;
}
<div class="my_div">
  <h1 class="title">
        <span class="old-text">This is the old text</span>
        <span class="new-text">A wild text appears!</span>
     </h1>
</div>