<p><a href="https://en.wikipedia.org/wiki/Stereolithography">Stereolithography</a> produces parts by projecting ultraviolet light on the top of a vat of liquid photopolymer, causing it to harden.  <a href="https://en.wikipedia.org/wiki/Continuous_Liquid_Interface_Production">CLIP</a> produces parts by projecting ultraviolet light through the bottom of a vat of liquid photopolymer, causing it to harden.  This seems like a minor difference, yet CLIP is reportedly much faster (I've seen numbers as high as 100x).  Why is this?</p>

It's important to understand what specifically is being compared.
CLIP is much faster than bottom-up technologies that require a peel step between every layer.
For example, the Form1 galvo SLA printer tilts the resin vat to separate the transparent bottom from the print. That is, by far, the slowest part of SLA/DLP printing with most modern light sources. Where the speed comes in is that without a peel, a continuous "movie" can be used to cure the resin rather than a series of alternating images and peels.
Top-down printers can print dramatically faster than bottom-up-and-peel printers. CLIP is not necessarily faster than top-down. For example, the Gizmo 3D line of top-down printers are very similar in print speed to CLIP. (
http://www.gizmo3dprinters.com.au/
)
Most "consumer" SLA printers these days use bottom-up-and-peel techniques, because this has some practical advantages over top-down printers:
Way less resin is required to fill the printer when the part is pulled out as it builds rather than being lowered into the tank (along with the Z stage) as it builds. Resin is expensive. This also means bottom up printers can be smaller and have fewer mechanical parts such as leveling devices submerged in resin.
Standard resins contain an inhibitor chemical that prevents polymerization in the presence of oxygen, which causes the surface layer exposed to air (and low-level stray light) to not cure. So top-down printers must shoot light through a non-curing layer before reaching curable resin. This makes the tuning more sensitive and can somewhat reduce detail compared to a bottom-up printer curing right on the window.
Replacement vats or windows for bottom-up printers may be seen by manufacturers as a profit-generating consumable, since they have to be replaced somewhat frequently.
Top-down printers have to worry somewhat more about resin flow rates as the part is lowered. Air bubbles may be pulled into the resin or the fresh resin layer above the part may vary significantly in thickness if the part is submerged too fast for the resin viscosity. (Admittedly, bottom-up printers will experience excessive suction forces and potentially break off bits of the print at high peel speeds.)
CLIP is a bottom-up technique that doesn't require a peel step, because the vat creates an oxygen layer over the window that keeps the resin from curing directly on the surface and sticking. In that way, it arguably performs more like a top-down printer than a bottom-up printer.
Top-down printers that are designed to overcome the above issues and use high-intensity light sources can achieve exceptionally high print speeds. This includes similar "continuous" build techniques used as in CLIP.