You can find a list of services and merchants that accept Litecoin on the Litecoin-Wiki.
That said, Litecoin has only minimal differences to Bitcoin: It uses Scrypt as its Mining algorithm, and reduced the block interval to 2.5 minutes, while keeping the halving interval (by requiring four times the blocks before the halving).
TL;DR: Unless it is financially profitable to you, there seems to be little reason to mine Litecoin. You find my point-by-point reasoning below.
- First confirmation can be achieved in approximately 2.5 minutes.
This is the biggest advantage over Bitcoin, however, it is still too long for a transaction in a physical store, and in most other scenarios waiting for 2.5 minutes or ten minutes is hardly a difference.
- Litecoin can handle about four times the amount of transactions than Bitcoin can.
However, Litecoin only has only about 5,000 transactions per day, while Bitcoin has more than 100,000 (recently peaking 200,000 due to the spam-attacks), rendering this advantage irrelevant so far.
- Wallets have to store four times as many blocks.
A similarly utilized blockchain would be significantly bigger, and the additional blocks are especially a problem for SPV-wallets, which have to store four times as many block headers.
- Shorter blocks favor large mining pools and cause more SPV mining.
The recently more frequent SPV mined blocks (essentially empty blocks) have caused an additional upper bound on transaction capacity in Bitcoin, leading to discussion on whether even 10 minutes would be too short, if one were to start Bitcoin from scratch.
Scrypt vs SHA-256
- "Scrypt is GPU and ASIC-resistant."
While Scrypt benefits less from raw computing power, but also requires more RAM, it has reduced the advantage of GPU and ASICs slightly, but its resistance has since been disproven by the emergence of Scrypt ASICs. Essentially, you just had to buy different hardware in a similar game.
- Scrypt is harder to verify.
As the computation per Scrypt hash is more complex, it takes longer to verify blocks. This hampers the synchronization of normal nodes, and demands hardware wallets to have significantly more power in order to even keep up with the network. This limits the ASIC-resistance of Scrypt, as higher memory requirements would also cause slower verification.
- Scrypt does not offer a greater level of security.
In the end, the balance is found between the economic incentive, and the cost of providing mining power. The trade-off is the same as with Bitcoin, thus not providing any significant advantage.
Currencies are a means for society to measure individuals' contributions and claims on society. This is a network that strongly benefits from market penetration: Just like a phone is useful if you and your best friend have one, it is exponentially more useful if everyone has one.
Systems that display such a network effect show a strong tendency towards one dominant exhibit per society. Examples of the effect at work are the elimination of languages through globalization, Facebook vs Myspace, or E-Mail vs Snail Mail. In the case of cryptocurrencies, the internet is the scope of society this competition presents in.
Other than in previous examples such as Facebook, the language barrier is hardly an obstacle, as currency is its own language transcending cultural borders.
Due to Litecoin's similarity to Bitcoin, I see two long-term possibilities:
- Bitcoin thrives: In that case, there will emerge one (or very few distinctly specialized) dominant cryptocurrency. As Litecoin doesn't seem to feature any convincing advantages over Bitcoin's headstart in market penetration, I fail to see that Litecoin will at any point become more useful or valuable than Bitcoin.
- Bitcoin fails: Being so similar, I fail to perceive any characteristics that would enable Litecoin to succeed where Bitcoin failed.
"Transaction volume will scale through altcoins"
I remain unconvinced that anybody would be interested in holding and transacting in a multitude of currencies each and every day. It is inconvenient and confusing.
There may several cryptocurrencies catering to distinctly different needs and/or societies. However, it is likely that they would be coupled interchangeably in some fashion, such as perhaps the Bitcoin blockchain and the Lightning Network.
Litecoin has utility as a higher stake Testnet for Bitcoin
Litecoin has been used as a testing ground for new Bitcoin features in the past, for example when Litecoin activated segwit before Bitcoin. This is a benefit to Bitcoin users mostly, but I don't see how that drives demand for Litecoin.
Conclusion: There is little reason to mine Litecoin
Unless you find that it to be profitable to mine Litecoin and sell it to the greater fool, I find no other reason to mine Litecoin.