Well, the differences start when we consider what each sort of messenger really does in the cell. Also note that there are many, many more receptors and channels than what you mentioned here! Let's just do a quick summary of each of those things, and then we'll compare/contrast them.
Ligand-gated channels are channels that are opened on the binding of a substrate (ligand) to the channel. It's almost like a keycard that needs to be swiped to open a door. One of the best examples of this is the nicotinic receptor for acetylcholine (ACh). When nerves send signals, they use ion channels to let the signal travel. For one nerve to tell another nerve to start communicating, the first nerve needs to open the second nerve's ion channels. This effect is accomplished by using ACh to open the ligand-gated channels at the synapse between the two nerves.
G-Protein Coupled Receptors (GPCR's) work by binding a ligand and then releasing a molecule (that started attached to the receptor) to carry out one of a few side reactions. These side reactions then inhibit or promote certain activities in the cell. For example, in the heart, norepinepherine binds its GPCR, which then activates adenylate cyclase to make cAMP, which then activates a protein kinase, which then does a few things, culminating in fight-or-flight effects like increased contraction strength. A way to think of this is like setting off dynamite in a cartoon. Your ligand pushes on the handle, and the G-protein (in the box) sends the signal to the dynamite (other things in the cell) to explode!
The molecules involved in those side reactions are actually what is termed "second messengers." For example, in the particular GCPR we discussed, the second messenger would be cAMP, which activates the other molecules in the cascade. In a way, it acts like the Postal Service. You (the ligand) give a letter to the courier (activate the GPCR and send out your second messenger). The courier then brings the letter and signals whomever you sent it to.
Now, hopefully that it is clear that most of the similarities and differences here have to do with things being inside and outside the cell. With both ligand-gated ion channels and GPCR's, the ligand remains outside the cell. However, ligand-gated channels allow transport of something (ions) across the membrane, while GPCR's simply change shape to throw off a molcule that does the signaling. Nothing is moved across the membrane with G-proteins!
Now to relate these to second messengers. Ligand-gated ion channels have no second messengers. GPCR's always work with second messengers, either stimulating or inhibiting them. However, not all second messengers involve G-proteins! Tyrosine kinases are a great example of non GPCR second messenger systems.
A common misconception, by the way, is that G Proteins are channels of some sort. They might activate one through a second messenger. However, they by themselves are not membrane channels.
I hope that helps!