What is the difference between the discount window and Fed repos? I'm wondering how the Fed's discount window differs from it's involvement in the repo market.

To the best of my knowledge, both are tools used to conduct monetary policy and both are also short-term, collateralised loans for commercial banks that are facing liquidity problems. So what is the difference between the two? The only one I can think of is that the discount window has different tiers of loans whereas Fed repurchase agreements do not but this seems like too small a difference to be the only one between the two.
Economically, they are both loans against financial instruments. The difference is the collateral posted - repos by the Fed are generally against Treasury securities. This is not a small difference: it makes it much easier for a bank to access liquidity, since not all of its assets are Treasury securities.

There’s a mechanical difference: a discount window operation is borrowing, while a repurchase agreement is matched pair of buy/sell orders. 

Historically, there’s a big difference: banks only went to the discount window at the Fed if they faced liquidity problems. I.e., the Fed is the lender of last resort. This was greatly discouraged, as it was a signal that the markets did not consider the bank to be viable, and the regulators would want to know why. Since the Financial Crisis, there have been efforts to remove that stigma.