When the first day of Rosh HaShana falls in Shabbos (which is the only day of Rosh HaShana that can fall on Shabbos nowadays), we don't blow shofar; as discussed in a previous TfdD. Similarly, when Shushan Purim falls on Shabbos (which is the only day of Purim that can fall on Shabbos nowadays), then those in a walled city from the time of Yehoshu'ah bin Nun would read M'gilas Esther one day earlier (the same day as the rest of the world); as not discussed in a previous TfdD. The underlying (revealed) reason is a concern that someone would come to carry on Shabbos; the shofar on Rosh HaShana, the m'gila on Purim.
On the other hand, if the eighth day of a healthy, full term, born naturally Jewish boy's life falls on Shabbos then we do give him his Bris Mila on Shabbos. Chazal were apparently not concerned that someone might come to carry a knife (or the baby) through a r'shus harabim to get the job done. Why the difference?
The Dirshu Mishna Brura brings three and a half explanations. (The half is a reference to a L'vush that gives a fourth explanation; I am lazy and spoiled, so I'll probably wait till the Dirshu Addendum comes out.)
Tosafos (Megila 4b, dh v'ya'avirenu) suggest that the mitzvah of bris mila is just way too severe to worry that anyone would make such a mistake. After all, the bris mila comes with 13 covenants. Besides, while anyone can blow a shofar (I've even done it) and read m'gila, only an expert is going to perform a bris mila. It is just not relevant to worry that he'll be an expert on bris mila and at the same time so unknowledgeable about Shabbos.
The Rahn (Rosh HaShana 8a) says that the only time you have to worry about someone forgetting themselves and carrying for a mitzvah is when everyone is involved in that same mitzvah. With everyone involved and concentrating on their mitzvah observance, they might not notice someone about to carry. That reason just doesn't apply to the mitzvah of bris mila, which has only a few people running around like chickens with their heads cut off. (Another of my favorite expressions.)
Finally, the Ritvah (Megila 4b) says that the difference is because of how the time to execute the mitzvah is specified. For bris mila, the time of execution (perhaps a poor choice of words...) is on the eighth day; not much room for error. Shofar and k'ri'as megila, on the other hand, are specified by a date on the calendar. Calendar dates, as we all know from second day Yom Tov, can be off by a day. It would therefore be possible that someone would be carrying the shofar or megila on the wrong day and therefore being m'chalel Shabbos for no good reason.
Whatever the reason, the message Chazal sent is loud and clear: it's just not worth even millions of Jews performing a mitzvah if there is the slightest possibility that one Jew will come to sin -- even accidentally.