Whew!
I'll try in a very non-technical way to explain.
The Breguet Spring is actually a hairspring bent with an overcoil at one end. It is intended to compensate, partially, for "isochronism error", which is the error caused by a difference in force as a spring unwinds. If you think about it, a spring has much more power initially unwinding than at the end. That difference can affect timekeeping (consistency). Most watches don't need a Breguet overcoil today, since modern metals have different coefficients of elasticity --but it can make a difference, especially with large balance or watches with lengthy power reserves. Breguet invented the overcoil technique over 200 years ago, and today all but one company has the bending done by hand, which is highly-skilled artisanship. An expensive and nice addition to a watch.
A Faraday or soft-iron cage shields the watch movement from magnetic impulses. It has nothing to do with the Breguet overcoil. It is nice, but most watches today for normal use survive just fine without a soft-iron cage.
A Glucydur beryllium alloy balance is the wheel that revolves, as the hairspring unwinds, and initiates the timekeeping process. Today, 99.9% of fine watches have balances made of this alloy. The alloy has a low coefficient to temperature changes, and helps avoid the timekeeping consistency being very far off as the watch is exposed to normal changes in heat and cold.
The high precision adjustment cam on the balance arms, however, is a big deal --balances sometimes run a little fast or slow (sometimes due to a knock, etc.) and need to be regulated. The traditional way is by having a "regulator" index, which when moved left or right shortens or lengthens the hairspring. The better way is by cams on the balance or arms with act like mini-weights, and change the inertia of the balance. That is more complicated to adjust but those seldom go out a whack, unlike a spring regulator. Such balances with cam adjustment often are called "free sprung" balances.
By the way, all the above are reasons that IWC Calibres 5xxxx, as used in the Big Pilot, the Portuguese Automatic, etc., are both costly and really fine movements. And beyond just timekeeping or avoiding repair, there is a real craft to the assembly of these movements --and to me the name of the game is craft.
