Due to the lack of adequate answers, I decided to research the differences myself and provide an answer to the benefit of the community.
PAR Type Lamps
From a build quality and light control standpoint, PAR type lamps are generally considered superior. The explicit parabolic nature of the reflector means light is more precisely reflected directly out of the bulb cavity, with less dispersion, than an R or BR type lens. PAR type lamps, thanks to the design of the lens in front of the actual light emitting element (which is usually a given with home lighting...PAR in general can referr to a very wide variety of lighting which may or may not have a front lens, including high powered stage lighting). The lens is often fresnel in nature, are often additionally capable of focusing the more accurately reflected light into a brighter, narrower spot. The fact that they have a more precisely crafted lens means they are capable of being designed to emit beams of varying angular degrees wide (anywhere from 12° to 70°), with high intensity narrow beams or more diffuse wider beams, makes them highly flexible.
PAR type lamps are also frequently designed in such a way that makes them viable for outdoor use. Not all PAR type lamps are properly sealed for use in humid climates or areas where rain or other water could intrude upon the bulb, but many PAR type lamps are. This is a key difference between PAR and R or BR type lamps.
In terms of nomenclature, PAR lamps are usually designated with a number. The number of a PAR lamp, such as PAR38, gives the diameter of the lamp in inches. In the case of a PAR38, the diameter would be 4.75" (4 3/4"), or "thirty eight eighths of an inch." Common PAR lamp sizes for home are PAR20, PAR30, PAR38, with the latter being most common.
R Type Lamps
One of the more common types of flood light lamps for use indoors, the R-type lamp is a more cheaply constructed "reflector" type lamp. They are not particularly efficient in any key respect, their design is roughly defined and there are no strict rules or guidelines regarding the reflector. As such, R-type lamps generally produce a highly diffuse light with a broad angle.
They use a similar nomenclature as PAR type bulbs. For example, an R40 is a bulb 40/8ths of an inch in diameter, or 5". The two most common R-type lamp sizes are R40 and R30.
BR Type Lamps
Preceding the CFL and LED age of high efficiency lighting, the BR-type lamp was an attempt to produce a more efficient indoor-use only replacement for R-type lamps. Standing for "bulbous reflector", the design of a BR-type lamp is more strict. The reflector is designed in such a way as to more efficiently reflect light. They still produce fairly soft, diffuse light, however it is reflected in a narrower beam, thus more efficiently utilizing light produced by the internal emitter (filament, LED, or CFL spiral.)
They too use the same eighth-inch nomenclature as PAR and R type lamps. The most common sizes are BR40 and BR30.
CFL and LED R, BR, and PAR type lamps
While the core design of these types of lamps was implemented in the days of tungsten-filament (incandescent and halogen) bulbs, the design is still used for CFL and LED lamps. In the case of CFLs, the actual reflector efficiency is likely suspect (at least in the case of a PAR type lamp), as the design of a PAR lamp, and thus it's efficiency in utilizing light, is dependent on the size and location of the internal emitters. CFL spirals are usually used in CFL-based flood lamps, and CFL spirals are a very different kind of emitter than a filament or LED. LED based flood lamps probably resemble classic filament based flood lamps better than CFLs do.
LED and CFL Light Quality
In some experimentation of my own, LED-based PAR type lamps definitely seem to direct a far greater quantity of their light downward, where as CFL spiral-based lamps still generally produce a more diffused output. Depending on the type of lens in front of the emitter, the quality and shape of the projected light for both CFL and LED PAR type lamps can vary greatly, from caustic refraction to fairly diffuse. CFL R/BR type lamps nicely simulate their incandescent predecessors. LED R/BR type lamps do not seem to diffuse quite as well as CFLs, so if you are looking for diffuse flood lighting, CFL generally has a more pleasing diffusion.
LED PAR lamps definitely produce a greater amount of directed illumination, so if you need to brightly light anything, LED is probably the best choice. LED flood lamps are also usually fully dimmable, "instant" on (sometimes "instant" really means up to a half-second delay before light is actually emitted), and produce far greater consistency of color (CFL color consistency is often rather poor, and during cooler or cold days, CFL floods will frequently brighten, dim to a deep magenta, then slowly normalize as they warm). LED lamps, of all designs, are usually fully dimmable these days as well. Dimmable LED flood lamps are usually dimmable down to 10% illumination, with some from the better manufacturers (such as Lighting Science) often dimmable down to 5%. Dimming quality, smoothness, minimum output and consistency is vastly superior on LED compared to CFL, with no flicker, popping, blinking, or inconsistent dropout as is frequent with dimmable CFL lamps.
In terms of the quality of projected light, BR-type CFL lamps definitely produce a smoothly diffuse light with little visible pattern directly underneath each bulb. PAR type lamps, either CFL or LED, but more so in the case of LED, produce visible refraction patterns under each bulb. I believe this has a lot to do with the design of the lens, and when multiple flood lamps are used along a hall or in a kitchen, diffusion improves. LED PAR type lamps can be very bright, which is nice when you need light.
My Choice and Recommendation
I have chosen to refurnish my kitchen, and eventually my hallways, with Lighting Science 5000k and 3000k (or possibly 4000k) Dimmable PAR38 LED lamps. These puppies are about $34 each, however they have twice the lifetime rating of comparable lamps from all other manufacturers (50,000 hours @ 6hrs per day vs. 25,000 hours @3-6hrs per day). The light is not as diffuse as your standard R-type incandescent, but the color quality is far superior for those who have never much liked the deep orange color of a classic incandescent (the 3000k versions are closer to a halogen in color, and the 5000k produce a nice, clean, crisp neutral white.)
For the highest quality, I recommend the following. Since LED lamps are dimmable, it is recommended to use a dimming switch wherever you use high lumen LED PAR lamps (i.e. 800 lumen or brighter). Personally I use Lutron HED-certified digital dimmers with dimming setting memory. One push of the button and the lights fade up over about a 1.5 seconds to your pre-set brightness level. Another push of the button, and the lights fade to minimum brightness then off over several seconds. With 800 lumen bulbs, setting a Lutron digital dimmer one or two notches down from maximum brightness produces very pleasing illumination, with the added bonus that if you need more light, or wish to add light during the daytime to fully illuminate areas that might generally be in shadow, you have the ability to crank up the illumination beyond that pleasing level.
Best Answer
It sounds like the line voltage is low. I have had this issue before with some cheaper and counterfeit bulbs from China. Check your line voltage at the socket (the thing the bulb screws into) and see how much of a drop it has from, say, a nearby outlet. You can then get a general idea of how much voltage drop the 4 LED bulbs are causing in total. What happens is that the little inverter type power supply board in the LED is really fussy in the cheapies, and if the voltage at the junction box is already low (old house, end of a trunk , other loads in line, etc.), the voltage is too low to run five, but just enough to run 4. you can sometimes see this if you take a high frame rate HD video (70fps and higher) of 1 led bulb installed, and then with 1 less than the number that is stops working at (5 in your case, so 4 for you). Then playback the videos on a player at slow speed. You will see a flicker with 4, but no flicker with just 1. T322his is due to the driver voltage being just sufficient with 4 lights, but too low with 5.
If you have a dimmer, try all 5 without the dimmer. If that works, you either need a magnetic driver type dimmer or you have non dimmable bulbs. If it doesn't work, your line voltage is too low and it simply won't work. And remember, this is all assuming you have checked for shorts, bridging, etc in the fixture, wiring and sockets.