Edit in JSFiddle

Result
HTML
JavaScript

<body>
  <div class="wrapper">


  </div>
</body>

const createDOMFromSring = domstring => {
  const div = document.createElement("div");
  div.innerHTML = domstring;
  return div;
};

class UpVoteButton {
  constructor() {
    this.state = { isUpVoted: false };
  }

  setState(state) {
    const oldElem = this.elem;
    this.state = state;
    this.elem = this.render();
    if (this.onStateChange) this.onStateChange(oldElem, this.elem);
  }
  changeUpVotedText() {
    this.setState({
      isUpVoted: !this.state.isUpVoted
    });
  }

  render() {
    this.elem = createDOMFromSring(`
      <button class='upvote-btn'>
      <span class='upvote-text'>${
        this.state.isUpVoted ? "Cancel" : "UpVote"
      }</span>
      <span><i class="fa fa-thumbs-up" aria-hidden="true"></i></span>
      </button>
    `);
    this.elem.addEventListener("click", this.changeUpVotedText.bind(this));
    return this.elem;
  }
}

class DownVoteButton {
  constructor() {
    this.state = {
      isDownVoted: false
    };
  }

  setState(state) {
    const oldElem = this.elem;
    this.state = state;
    this.elem = this.render();
    if (this.onStateChange) this.onStateChange(oldElem, this.elem);
  }

  changeDownVotedText() {
    this.setState({
      isDownVoted: !this.state.isDownVoted
    });
  }
  render() {
    this.elem = createDOMFromSring(`
      <button class="downvote-btn">
      <span class="downvote-text">${
        this.state.isDownVoted ? "Cancel" : "Downvote"
      }</span>
      <span><i class="fa fa-thumbs-down" aria-hidden="true"></i></span>
      </button>
      `);
    this.elem.addEventListener("click", this.changeDownVotedText.bind(this));
    return this.elem;
  }
}

// as render() method now returns an Object of HTMLDivElement
// it cannot be appended like innerHTML, instead use DOM
// API to append it
const wrapper = document.querySelector(".wrapper");
const upVoteButton = new UpVoteButton();
wrapper.appendChild(upVoteButton.render());
upVoteButton.onStateChange = (oldElem, newElem) => {
  wrapper.insertBefore(newElem, oldElem);
  wrapper.removeChild(oldElem);
};

const downVoteButton = new DownVoteButton();
wrapper.appendChild(downVoteButton.render());
downVoteButton.onStateChange = (oldElem, newElem) => {
  wrapper.insertBefore(newElem, oldElem);
  wrapper.removeChild(oldElem);
};