The smaller bullet in the photo is definitely from long after the end of the civil war. Enlarging your photo shows the small bullet has multiple tiny parallel ridges inside the groove which encircles the bullet's body. That is what is called a "reeded groove" or "knurled canellure"... and that kind of body-groove was not used on bullets until 1877. See the photo below for a closer view showing the tiny parallel ridges inside the groove on a 20th-Century bullet..