original.js

function getSomeThings(callback) {
  var completed = 0;
  var people, tasks;

  $.ajax('/data/people.json', {
    dataType: 'json',
    success: function(data) {
      completed++;
      people = data.people;
      onFinished();
    }
  });

  $.ajax('/data/tasks.json', {
    dataType: 'json',
    success: function(data) {
      completed++;
      tasks = data.tasks;
      onFinished();
    }
  });

  function onFinished() {
    if (completed !== 2) { return; }
    callback(people, tasks);
  }
}

refactored.js

function getSomeThings(callback) {
  var peopleRequest = $.getJSON('/data/people.json').then(function(data) {
    return data.people;
  });

  var tasksRequest = $.getJSON('/data/tasks.json').then(function(data) {
    return data.tasks;
  });

  $.when(peopleRequest, tasksRequest).then(callback);
}

function getSomeThings(callback) {
  var reqs = $.map([ 'people', 'tasks' ], function(idx, req) {
    return $.getJSON('/data/' + req + '.json').then(function(data) {
      return data[req];
    });
  });

  $.when.apply($, reqs).then(callback);
}

+1 to what @rmurphey said. This is the difference between refactoring and rewriting. Refactoring doesn't change your external API, it only changes the internals. Rewrites have the freedom to change the API.

Agreed. I didn't know the context of the code samples.

I'd still return the $.when though, so future consumers can get the benefits of deferreds.