I don’t think LN is very much linked to ElementsProject. Somehow they have similar intent: scaling bitcoin. However they do it in very different ways.
Elements is a federated sidechain, a somehow complete blockchain that bases parts of its security to a federation of entities that create blocks that pegs to the bitcoin blockchain every now and then. Being a separate blockchain, it can support different things than bitcoin: confidential transactions, assets, simplicity script, etc. Moreover, since blocks are created by the federation and there is no mining, blocktime can be a lot shorter (~1s), which leads to low fees and fast settlement.
Lightning Network instead does not rely on somewhat trusted entities to scale bitcoin. It aims to build a payment channel network as a layer on top of bitcoin, relying on it just for the settlement of controversies. It works somehow well, but it brings in quite a bit of complexity and it changes a bit the security model we were used with on the base layer. To keep funds safe, a LN node has to be “always” online or to delegate to external entities (called watchtowers) the job to keep an eye on the blockchain to protect your funds.
Interestingly, there is a nice project under development that aims to bling most of Element’s specific functionality on top of bare bitcoin and LN through the concept of client-side validation. You can find something more on RGB here and here
