The basic story is that a goy approached Shamai and Hillel to convert if they could tell him the basic meaning of Judaism "on one foot". Shamai, having the impression that the goy was being sarcastic (as I interpret the gemoro) and asking for the impossible, chased him away with a "builders measure". That is Shammai saw that he was not sincere and chased him away with something that symbolized Hashem created the universe with mishpat. Hillel gave his famous statement (ending with "now go and study") which convinced the goy to actually convert properly and study.
@CraigFeinstein pointed to this
Shammai, feeling that he wasn't serious, chased him away.
The Kli Yakar explains that this gentile was not mocking the Rabbis at
all but was, in fact, being very sincere. He, who hadn't studied Torah
from a young age, was looking for a single foundation upon which he
could base and thereby remember the entire Torah.
The point is that a general philosophical statement does not lead to proper understanding until an actual practical understanding has been reached. The way Hillel expressed it showed the goy that one needs to understand what one is doing and have a specific method of carrying it out.
What does it mean "Love your neighbor as yourself" and what practical means are there to actually carry it out? Hillel's formulation is designed to allow someone who has not studied Halacha to be able to start with concepts that he can understand and have a practical means of carrying it out without getting into meaningless philosophical arguments. One can say that many of the prohibitions of the Torah are formulated this way for similar reasons.
We can see this by examining the history of those who say "Love your neighbor as yourself" and never actually create laws and procedures to carry it out properly. They acted like the modern leftists that base everything on "feelings" and not on actions. Note that Hillel does not speak of feelings (he does not say "do not hate") but instead points out a specific way of acting. That is, those things that you would find "hateful" or wrong if someone else does it to you should not be done to others. This allows people who live in a society to begin to understand their responsibilities rather than concentrating on the "rights" that other people must grant them.
The prohibition against stealing shows the responsibility to ensure that others are "secure in their persons and property". On the other hand, the right to "be secure in one's own person and property" has not prevented people from stealing while at the same time enforcing their own rights.