Fist of all, you should clarify whether you are talking about just CPUs, GPUs or both. Assuming you mean GPUs, I'm an AMD guy, but here are the advantages and disadvantages of each brand:
For CPUs:
AMD
Advantages:
-Cheap (value for money)
-High frequencies (GHz)
-Multi-cores (4-8 average)
-Large L1/L2/L3 Caches
-Work extremely well with AMD GPUs, especially the APU families
-Driver support is instant! Bug fixes are addressed in less than an hour!
-Almost all AMD CPUs have unlocked clocks, meaning it's incredibly easy to overclock them!
-Overclock tolerance is insanely high! You can easily reach twice the initial frequency with proper cooling!
-Backwards socket-chipset compatibility! (eg: AM3+ sockets support AM3+ CPUs AND AM3 ones.)
-APUs only: They consume/drain far lower wattage, resulting in lower electric bills.
Disadvantages:
-In all not APU families, cores work collectively instead of independently, meaning it's like you have just one HUGELY powerful core, making heavy multitasking slower and may even hang under full load.
-In relation to the above point, core management is terrible and sometimes even random!
-Without proper cooling the temperature can shoot up a LOT!
-Mother coolers (those are the ones provided when you get the CPU) are really loud! Even at IDLE state!
-The 8-core FX series, meaning 8320-9990 don't include a cooling device/system and need extremely powerful cooling to keep them from overheating!
-Component support is greatly lagging behind compared to Intel. For example, AMD has yet to commercially release a CPU for use with he DDR4 rams, while Intel did it on their launch date back in January 5.
Intel
Advantages:
-Core management is the second best in processors there is! Meaning great multi-core support!
-Slightly higher performance than their AMD equivalents (just 4% though).
-Greater multi-system support (same core can serve in any kind of system
-Instant hardware support as soon as it's released! (example above^)
-Better quality heatpipes, so better heat circulation, meaning that they don't reach high temperatures that easily.
Disadvantages:
-Price! They cost almost THREE times as much as the AMD counterparts! (remember, just for 4% more performance!)
-Driver support takes FOREVER! AMD released a bug fix for processing errors in Batman Arkham Knight on the first day, while Intel has yet to even start working on one!
-WAY too many sockets! The same processor may come out for a different socket, without compatibility between the different versions and without an significant differences!
-When not using multi-core applications, the performance is at ~35% of that of AMD processors!
-Has way too many disadvantages to keep listing them!

(Also, has less advantages than disadvantages)
There is no need to make a list about the GPUs, as AMD's are the best in performance-price ratio and second best in sheer performance, so I'm leaving it at that.
If you need any more information, just ask.