Siam Square is the most popular shopping area in Bangkok. In this area, there are a wide range of shops and services such as restaurants, coffee shops, banks, boutique stores, bookstores, and tutor schools. It is a meeting place of high school students, college students, and also office workers. Besides, this area is close to many shopping malls around such as Siam Square One, Siam Paragon, Siam Center, and Siam Discovery.
Chinatown (or Yaowarat) is a place of Thai-Chinese community. During the day, it is a market that you can buy foods and ingredients. At night, it is a well-known place for street food.
Siam Square connects with the BTS Skytrain, while currently Chinatown does not connect with any express train. The nearest MRT station to Chinatown is Hualamphong station; after getting off at this station, you have to walk around 20 minutes from there. There will be a MRT station in the middle of Chinatown soon, but it is now under construction. If you plan to visit Siam Square and then go to Chinatown, you might have to take either bus or taxi.
The distance between Siam Square and Chinatown is about 4-5 kilometers. It should take around 15 minutes to half an hour and should cost around 60 baht, if the traffic is not really bad. It can take more time and money during peak hours such as 6.00-9.00 or 17.00-20.00. So, try to avoid these periods, if it is possible.
You can take bus no. 25 and 40 at the bus stop opposite Siam Paragon Shopping Mall. This option is cheaper, but I do not recommend if you are not familiar with Bangkok.
Hope this information help