What would be the consequence of a constant network difficulty? Say the difficulty of the bitcoin network was not variable. Then regardless of the network hash-rate, the mining puzzle will be the same. What would that mean in terms of the security and economics of the bitcoin network?

If everything else remains equal, then I'm assuming that mining a block becomes easier and easier and we will have mined all 21 million bitcoins much sooner. Besides this, will anything change security wise? And if so, can anything be done to maintain the decentralised consensus and fiscal policy of the network with a constant difficulty? For example, the block reward payed to the miner becomes inversely proportional to the number of parent blocks.
You would eventually be up against network propagation speed, since miners would get powerful enough to mine faster that the blocks can propagate across the network. The nodes with the lowest latency between each other would always be ahead with the longest chain, penalizing those with higher latency.

You're correct that the 21 million will be reached much sooner.

You can observe this on the testnet when the block times are high enough, after it resets to the initial difficulty and blocks come flying in multiple times a second (basically as fast as the network can handle it). For more info see Testnet difficulty change