Both quantile normalization and factor elimination can be seen as a kind of normalization, but they are used in different ways. The quantile normalization is usually used to normalize the entire set of observed values in order to make different samples comparable. It works by forcing the distributions of values for different samples to be identical. Hence, after quantile normalization the collection of values (for all variables) will be the same for each sample, they will just be arranged in different orders (so that the most highly expressed variable in a specific sample obtains the highest value, etc). The factor elimination, on the other hand, works on each variable individually (whereas the quantile normalization works on each sample individually). Also, it is less "invasive" than the quantile normalization in the sense that we only make the mean values identical across the sample groups (in the simplest case with a single nominal factor) instead of the entire distributions. So conceptually, both meancentering and quantile normalization can be seen as normalization methods for making different values comparable (give them a common "baseline"), but as described above they are used in quite different ways.
