Native mode is more technical and it requires you to run a full node (download the entire blockchain). The arguments in favour of “most users should be running a full node” can be broadly categorized into two groups: a) its good for the network, and b) its necessary for the security of the user. An example of native wallet is the Bitcoin Core QT wallet.
Lite mode is less technical and easier to use. It is for users who are on the go and don't necessarily need to have a full node. In a nutshell, SPV lets you validate YOUR transactions without having to worry about anybody else’s transactions. Lite mode (SPV) provides the 2 critical ingredients: a) It ensures your transactions are in a block, and b) it provides confirmations (proof of work) that additional blocks are being added to the chain. An example of a Lite wallet is the Electrum SPV client.