Do Monero and ZCash have different transaction fees?

This is largely going to depend on the price, which is currently unknown because Zcash hasn't launched yet. However, on testnet the fee seems to be approximately 0.00005 TAZ (Zcash tesnet coins) per kB, whereas Monero currnetly has a fee of 0.002 XMR per kB.

A more interesting comparison is transaction sizes. The average transaction size for Monero is currently approximately 1 to 3 kB. By contrast, the anonymous Zcash transactions seem to be approximately 2 kB, according to one of their engineers, namely Sean Bowe.

Do Monero and ZCash have different mining algorithms?

Yes. Monero uses CryptoNight as PoW algorithm, whereas Zcash uses Equihash as PoW algorithm. Both are aiming for ASIC resistance. However, one could argue that the CryptoNight algorithm is more vetted, because it has already been used in Monero for over two years. The Equihash algorithm, however, was until Zcash's implementation purely theoretical.

Do Monero and ZCash have different coin supply curves?

Yes. Zcash uses the same supply curve of Bitcoin, whereas Monero uses a faster supply curve. A comparison between both emission curves can be seen here.

There are two things to note. First, in Zcash 20% of the mining rewards will go to the Zcash Electric Coin Company, which is a US based company, in the first four years. This company subsequently distributes the so-called "Founders Reward" to its shareholders. These shareholders are founders, scientists, engineers, advisers, outside council, and investors. They can be seen here. In addition, the "Founders Reward" is depicted here. Secondly, in contrast with Monero, Zcash doesn't have a tail emission, which may lead to disincentivized miners in the future.