What is the difference between Amanita muscaria (Fly agaric) and Psilocybe mushrooms?

General differences

Psilocybin/Psilocin Mushrooms
Mushrooms that contain the hallucinogens psilocybin and/or psilocin belong mainly to the genera: Psilocybe, Conocybe, and Panaeolus. The word "psilocybin" comes from the Greek words "psilo" meaning "bald" and "cybe" meaning "head". Visually they appear usually with carmel/light tan to off-white caps of varying size and a white to brown stem possibly showing a bluish tinge.
Some specific mushrooms containing psilocybin and psilocin include Psilocybe cubensis, Psilocybe semilanceata, Psilocybe mexicana, Psilocybe pelliculosa, Panaeolus subbalteatus, Psilocybe cyanescens and Psilocybe baeocystis.

Amanita muscaria
The Amanita muscaria mushroom is also known as "Fly agaric" because of its (anecdotal) ability to attract and kill flies. The Amanita muscaria does not contain psilocybin or psilocin. Rather, the hallucinogenic chemicals this mushroom contain are muscimol and ibotenic acid.
The fly agaric is related to some deadly mushrooms: the Amanita virosa (the "Destroying Angel"); Amanita verna and Amanita phalloides (the "Death Cap"). These mushrooms contain toxins that destroy cells in the liver and kidneys. Five to 24 hours after eating one of these toxic mushrooms people may become sick with nausea and stomach problems. Later, severe liver and kidney damage may occur, often resulting in death.Visualy these are the classic "toadstool" with a bright red to orange cap covered in white flecks and a strong white stem and gills.

Effects on psyche and body

Psilocybin/Psilocin Mushrooms
At low doses, psilocybin causes simple feelings of relaxation, physical heaviness or lightness, and some perceptual distortions (especially visual). At higher doses, more physical sensations occur, including lightheadedness, oral numbness, shivering or sweating, nausea, and anxiety or hilarity. The symptoms may appear 30 to 60 minutes after the mushrooms are eaten and reach a peak level around 1.5hrs or less and can last about four to six hours with an "afterglow" lasting several hours to several days.

Amanita muscaria
Amanita muscaria contains muscimol that produces feelings of euphoria, hallucinations, muscle jerks, drowsiness, sweating, pupil dilation, Increased tear and mucus formation (Those on muscaria often exhibit excessive tearing and salivating) and increased body temperature. Symptoms appear 30 to 90 minutes after eating this mushroom and are most intense after two or three hours. Symptoms include blurred vision, loss of motor skills, moderate to extreme nausea or cramps, physical relaxation or a sedative like effect, euphoria and a “dream like” state of being. The experience is said to be very different from that of psilocybin and many do not enjoy it. People who eat these mushrooms sometimes fall into a deep sleep. Some people describe the effects of eating Amanita muscaria as similar to being intoxicated by alcohol.

Effects on the Nervous System

Psilocybin/Psilocin Mushrooms
The chemical structure of psilocybin and psilocin is similar to the neurotransmitter called serotonin. In fact, the primary effect of psilocin is on the receptors for serotonin (HT-2a). There is also evidence that psilocybin reduces the re-uptake of serotonin by neurons in the brain allowing this neurotransmitter more time to act in the synapse.

Amanita muscaria
Muscimol and ibotenic acid from the Amanita muscaria appear to act on the GABA neurotransmitter system. Muscimol activates GABA receptors on neurons. The GABA neurotransmitter system is one of the brain's major inhibitory systems. Therefore, muscimol acts to inhibit the activity of neurons in the brain.

Sources:
Neuroscience for Kids - Hallucinogenic Mushrooms
Erowid Psychoactive Amanitas Vault
Miscellaneous Drug responses