One of the main differences is what the tamper is made out of (the casing). You need a material that is reflective to the X-rays used to compress the fusion fuel. In Castle Bravo, this was made out of uranium-238, in the Tsar bomb, it was made from lead. Using uranium boosted the power output by about 2x, at the expense of making the bombs far, far dirtier. The extreme heat and the high neutron radiation from the primary (a fission device) detonating would cause the uranium to decay into more unstable isotopes (U-235, Pu-239, etc.) which would then fission themselves.

The ideology at the time was that a larger blast was more important, since at the time nuclear weapons were still to be dropped by bombers (No ICBMs for another decade). You wanted the bomb to take out the target even if you couldn't quite reach it.

The Soviets estimated that using the same technique would have produced a yield of 100 Mt, but for testing, lead was used to limit the fallout to something they deemed manageable.