What do big data and high dimensional data mean? Is high dimensional data a special case of big data? What are the complications that arise in the analysis of high dimensional and big data each?
Big data implies large numbers of data points, while high-dimensional data implies many dimensions/variables/features/columns.
It's possible to have a dataset with many dimensions and few points, or many points with few dimensions. But if you have high-dimensional datasets with few data points, you're unlikely to be able to learn much from it. So high-dimensional data is generally going to be big data as well.
The converse is not true - big data does not need many dimensions for you to learn from it. But if you are only working with a few dimensions, it's probably not as necessary to collect a large number of data points to do your analysis. Note that there are important exceptions to this - noisy measurements, high frequency spatial or temporal data, and so on.
So it's probably generally true that giant datasets with many points also happen to have many variables/dimensions as well. In other words, the terms mean different things, but big data is usually high-dimensional data and vice versa.
Regarding the complications associated with each, here is a very incomplete answer: big data poses computational challenges (loading data in memory, for example), while analysis of high-dimensional data falls prey to the curse of dimensionality.