What makes Rinkeby a "testnet"? I understand it's a parallel network so my ether there has no effect on the mainnet ether. But how is it different technically?
Rinkeby is a Proof-of-Authority network, so uses a different consensus mechanism to the main net. The Ropsten testnet is Proof-of-Work, so more similar to the public main net.
But does it still run on miners?
Rinkeby doesn't, Ropsten does.
What incentive would miners have to maintain the testnet?
This doesn't apply to Rinkeby, but if we consider Ropsten: there's no real financial incentive, unless there's an elicit black market for testnet ether I've not heard about. (I wouldn't be surprised if there was.) Mining helps the developer community, so the incentive is the same as any other altruistic act.
Could people simply decide that Rinkeby ether is not that bad and start trading it as a currency in itself??
Yes, they could, but...
Does it reset itself periodically?
...yes. Rinkeby, being PoA, is run by centralised nodes which can be taken down whenever. (I'm uncertain who actually runs them. Anyone?)
Ropsten, being an open PoW network, is more decentralised. However, given the lower rate of hashing being used to mine the blocks, and therefore secure the network, it wouldn't take much to attack and/or ruin. (The original testnet, Morden, had to be taken down, as did the first iteration of Ropsten. We're currently on the Ropsten "Revival" network.)