One needs both parts of the description to identify himself as such. For socialism is a form of organizing the social system of production, namely, the idea that the state shall own all the factors of production, including, in this doctrine’s extreme variant, all labor. But socialism need not be egalitarian. A dictator or a committee of dictators can have all sorts of arbitrary powers over their subjects. They can have complete authority to kill them, to send them to prison or to life of hard labor, to determine their occupations and their family life (if such there be), to organize gladiatorial battles for the dictator’s amusement, to take any girl to bed and then have her executed the next day (the tale of Scheherazade being the familiar example), and numerous other outrageous things the imagination readily suggests. So, they would own everyone as their slaves, who would be miserable, while the dictators and their friends would live it up (to an extent).
On the other hand, egalitarianism is concerned with consumption, namely, that no one is to have more than his fellows to enjoy. And an egalitarian society need not be socialist. In fact, there is an argument for the assertion that a laissez-faire capitalist society becomes ever more equal. To reuse the example in a previous post, there is a bigger difference between a starving man and a sated man than between someone who has only one yacht and someone who has five, or between someone who flies coach and someone who flies first class. Under capitalism, there will always be a “trickle-down” process of luxuries being introduced to the innovators in the art of consumption (indeed, the rich), and after a fairly short while becoming necessities without which the common man could barely imagine his life. But with economic progress, as time goes on, each luxury first created will be ever more rarefied and seemingly “out there” and will improve the lives of the rich to a lesser and lesser extent as compared to the lives of the masses.
As an example, consider the Internet and sites such as Amazon.com and Google Books and so on. Knowledge of the truth about any subject, once the most precious, expensive, and worthy commodity one could ever purchase in his life, accessible only to a select few, has finally become cheap as dirt. Only princes and the aristocracy could at one time afford to hire scholars. The sky is the limit now; truth and debate about pretty much everything, created entirely by the market, is just a mouse click away. Gnosticism, understood as presumption of secret knowledge that perhaps entitled one to salvation or other goods, is dead and buried. That is how capitalism diminishes inequalities.