Downloading the full blockchain alone doesn't make one a full node. It performs additional functions such as verifying of signatures, check for the transaction if they are double spent, check that the newly created blocks are generating only certain number of btc etc. We need to note that full nodes are the ones, which gives miner the transactions and further these miners pick up those based on the txfee priority. A tx which pays high fee are mined faster than the tx with lower fee. If a full node doesn't feel like the transaction is legit and not following the consensus they have the power to invalidate the tx from getting mined. Miners just pick up the tx and add them in a block and mine them.

Whereas in a light node you need to download the block headers so that you don't need to depend on the full node at all times during a transaction. These block headers contain certain information in the form of Merkle tree.  This merkle tree contains information of a transaction hash and doesn't require you to have the whole body of the transactions for verifying.