In short, the answer is yes. There are differences between brain games and brain training.
It's very much like doing a couple of sit ups compared to going to the gym and following the regimen specified by your trainer. This does not mean that brain games do not have their uses. They do.
But to get the full benefits of optimal brain activity, you want to have a more rigorous program defined that keeps you on track.
Of course, it will not hurt to start with playing games that you love. Many people love to do crossword puzzles and do them without even giving it much thought that it could be helping their brains. If you believe that keeping your brain active will keep it healthier later in life, then doing crossword puzzles frequently is a good activity to engage in.
If you are not into them, you should consider puzzle games.
These types of games will get you to focus on problem solving. There is some belief that pathways created in our brain from the result of solving problems can help to reinforce the brain's capacity. (Note: Not all scientific studies are in agreement with this conclusion though.)
Do you tend to be more analytical?
Then, you may want to focus on quizzes and other games like trivia. These games can challenge your knowledge. Repeated use and practice of them will enhance your learning, at least in the subjects that you choose to quiz yourself on.
The last group of games that you could participate in are ones where you need to think ahead in order to solve the puzzle or problem at hand. Sudoku and chess are great examples of these types of games. Chess requires the projection of several moves in advance.
Brain training offers a more formal approach than brain games.
Here, you are given a curriculum and you are requested to perform brain exercises, including brain games. The training is often structured for each individual. The idea is for you to find the right approach that meets your individual needs.
Remember, the motives of many brain training companies are to make a profit.
Although companies have the right to make money, make sure the companies you are considering are legitimate. Ask a lot of questions and try to seek the advice of others who experienced these programs.
Otherwise, you may end up paying exorbitant amount of fees and receive little to no benefits to your brain. Even if you decide not to enroll in any program, if you enjoy playing games, feel free to participate in any free brain games that you may find. It's a relatively healthy way for leisure than most games anyway!