I`ve always been led to believe that lithium grease will take higher temperatures without softening and being `thrown off` reciprocating parts and also resists emulsifying (mixing and dilution by water) better. Brown grease is for general purpose sliding shafts, trunnions, ball joints etc etc and where it is generally lower temp applications.
I personally always use a lithium based molybdenum grease for CV and non sealed wheel bearings.
Copper grease is a copper base suspended within a compound. If the compound evaporates then a thin layer of copper is retained thereby protecting the surface (so good for bolt/nut applications or where mating faces meet ie: alloy wheels to metal hubs) but is a general lubricant and is not designed for bearing lubrication.
Edited to add below the info from the Castrol site (just checked in the garage having read it and its LM grease I use (I always thought the M was for Molybdenum but its not [8|])
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Castrol LM Grease
Lithium based, high melting point grease, specifically designed as a wheel bearing grease for use over a wide range of temperatures. Also suitable for general purpose applications
Castrol Red Rubber Grease
A special, rubber compatible grease for use on hydraulic brake and clutch components where hardening or swelling of rubber must be avoided.
Useful in assembly of rubber components for brake, clutch and suspension units
Castrol BNS Grease
A non-melting grease, designed primarily for use in anti-friction bearings operating in high temperature conditions. Ideally suited for competition use
Castrol Moly Grease
High melting point, lithium based grease containing molybdenum, providing dry lubrication under extreme operating conditions.
Ideally suited for the following applications; King pins and bushes, shackles and suspension trunnions, bevel worm and peg steering boxes
Castrol L/EP0
Lithium based, semi-fluid grease. Ideally suited for use in grease packed differentials and gearboxes requiring a semi-liquid, self-levelling grease
Castrol Graphited Grease
A soft, waterproof grease, developed for the lubrication of springs and chains. Based on a high viscosity oil with a low melting thickener, it contains colloidal graphite for additional load carrying.
The graphite prevents metal-to-metal contact under shock loading conditions
Castrol CL Grease
A soft, tacky, calcium based grease. Formulated from highly refined mineral oil incorporating an adhesion additive.
It has natural resistance to water and good protection against corrosion.
Primarily recommended for chassis lubrication but also suitable for swivels and shackles
Castrol Impervia MDX
Soft, tacky grease, containing additives which give it excellent adhesion to metals.
Suitable for high temperature, high load applications including shock loading conditions