I would say that Japan is a culture that fits this description. It has thoroughly Westernized in many ways, but it has not done so completely.
We all know that Japan has become quite Westernized. Japan's economy was until just recently the second largest in the world. It built that economic power on industrial power, which is a very Western thing historically. It has become a democratic country, also a Western thing.
At the same time, Japan has not abandoned its traditional culture. Though it is not as prevalent as before, Japanese are (of course this is not true of all Japanese) still very much more reserved and formal than most Westerners are. Japanese are still very much tied to their traditional foods. They still do very quintessentially Japanese things like cherry blossom-viewing and autumn leaf-viewing. In these ways, Japan is still quite distinctively Japanese even as it has become an industrial, capitalist democracy.