There two types of silver (and gold) in the market available: silver coin and silver round.
I am looking answers for this two questions.
What are the main difference between them, and advantages of each?
Coins are legal tender. They're authorized by governments and have a face value.
Rounds are simply coined pieces of metal minted by private manufacturers. They do not have any face value and are not legal tender.
Rounds are used to own metal, they have no value other than the value of the metal in them. Any premium you pay over the price of the metal is the mint's profit.
Coins are also used as bulions (i.e.: to own metal and create profits for the government), but many times coins have limited issue and become valuable because of the rarity, specific issues with a specific coin (mistakes, impurities, exclusive designs), etc. So they also may have some numismatic value (depends on the specific coin).
Coins also have the assurance of quality of the authorizing government (and fakes are dealt with by the law as forgery of coins is illegal and is a crime), rounds however do not enjoy such protection, and any one can mint them (only copyright/trademark protections apply, where the enforcement is by the owner and not the government).
Re the advantages - coins (if you pick the right ones...) appreciate much more than the metal. However, this is mostly in hindsight, and most of the "bulion" coins do not appreciate significantly beyond the price of the metal unless there's something else significant about them (first year of issue, high quality certification, etc).
Rounds on the other hand are cheaper (1 oz round will be significantly cheaper than 1 oz coin), and monitor more closely the price of the metal. It is unlikely for rounds to significantly deviate from the spot price (although this does happen occasionally, for specific designs or if a mint goes out of business).
littleadv gave a great answer, but neglected to mention one thing. Modern minted coins usually only contain a (high) percentage of a precious metal. For example pre-1965 quarters are 90% silver and 10% other, to maintain strength and durability. Rounds of silver bullion are usually .9999%, or fine, silver, which is considerably softer.