Nodes do not necessarily mine bitcoin but miners pick transactions from nodes' mempools. Miners will most likely pick the transaction paying higher fee rate and ignore the other one.
The chances of getting included into a block depends mostly on the level of propagation of each transaction, though.
Was actually referring to mining nodes. Just as said above by hosseinimr93 no two same transactions can be in a particular nodes mempool, so if the mining node actually gets to mine a block and the first transaction is in its mempool and it meets its fee rate criteria then it can be included in the block mined by that miner or say mining node. After the first transaction gets confirmed, the other one in another nodes mempool will definitely be invalidated. What I was emphasizing on is that both the old transaction and new transaction can be included in a block since a miner can pick just any transaction (although they use high fee as the priority) they want from there mempool to include in a block they miner.