A running wallet is a node in the network.
A miner is a node in the same network.
A masternode is a node in the same network.

A running wallet provides the network with a service that is helping other nodes to connect and sync to the network and broadcast transactions.
A miner provides the network with an additional service that is storing transactions into the blockchain.
A masternode provides the network with additional services that are (currently - more is to come) anonymization of coins and instant transactions.

Miners and masternodes are rewarded for their services, and both need to prove somehow that they are legit - miners do it by proof of work, and masternodes do it by proof of service and proof of 1000 DRK ownership.