Well, the main difference really is that A.I. now is being designed differently than before.
When we consider the computer, or A.I., that beat Kasparov, it was in its primitive stages. Much like a computer program it was designed with a specific purpose and with the help of many world-class chess players.
Newer A.I. is a combination of computer programming, cognitive psychology and neurosciences. Cognitive psychology is the study of cognition which is how we perceive things and learn from those perceptions and experiences. An example would be being presented with an entirely alien challenge to us and using our experience, observation and current knowledge to solve the problem.
To see that in action with Deep Mind's A.I., there's a video of them challenging it with video games as it provides a convenient means of quantifiable progress.
Here is a video of the game learning how to defeat an Atari game. You see how it learns that the most efficient and effective way is to attack the sides and then get it on top to let it do the work itself!