Also no expert, but the basics as I understand them:
Free-sprung balances vs. regulated balances reflect two different ways of adjusting the rate (fast/slow) of the oscillating timekeeper. Free-sprung have a fixed-length hairspring, and thus the rate is adjusted by changing the inertia of the balance itself by moving weights (screws and such) either out from the center (increasing inertia and slowing rate) or in towards the center (decreasing inertia and increasing rate). Regulated balances have a fixed-inertia balance (mostly – more on this later) and a variable-length hairspring. The hairspring’s effective length is changed by the regulator: as you move the index on the balance cock, a pair of curb pins slides along the hairspring: in toward the center shortens the spring and increases rate, and vice versa.
Many watches with regulators, especially these days, also have screw balances. Sometimes these are timing screws (ie, with slots) which can be used to change the balance inertia, and sometimes these are weight screws (no slots) that serve an aesthetic purpose and to some extent to move the mass of the balance out the rim as a stabilizing factor. On screw balance watches with a regulator, the timing screws can be used to make large changes in rate, and the regulator for fine changes.
For a free-sprung system, the main advantages are reported to be stability of rate, once properly timed (no regulator to knock about, as Steve said), and better inherent positional accuracy. This is because the hairspring, particularly when paired with a Breguet overcoil, is free to “breathe” concentrically as it contracts and expands with balance oscillation. This concentric breathing theoretically means than the watch’s rate changes less when put into different positions.
The main disadvantage to the free hairspring is the difficulty in regulation. Rate is changed only by fiddling with the balance, and any non-symmetric changes will throw the balance out of poise (like a wobbly bicycle wheel). The main disadvantage to the Breguet overcoil is its height: it adds something like 1mm to the balance assembly.
For a regulated system, the main advantages are the ease of regulation (move an index) and the fact that you never (almost) have to adjust the balance, so it stays poised. The main disadvantage is the distorting influence of the regulator pins on the breathing of the hairspring: the breathing is asymmetric, and thus there are inherent positional rate issues arising from gravity which must be addressed for high performance.
After all that, I am entirely with SteveG: both adjustment methods can and are employed in the highest quality watches and can be and are adjusted to the highest level of timekeeping performance.
Lastly, I believe that Lange has retained the swan’s-neck regulator on its free-sprung watches not a a rate adjuster, but rather to adjust the beat (impulse from the pallet fork to the balance should be symmetric in both directions of the fork swing) by centering the fork at rest between two guide posts.