What's the difference between BTC and BCH transaction capacity? If Bitcoin can process x transactions per second, how many transactions per second can Bitcoin Cash process?
tldr - BTC = 3.33-6.66tx/second, BCH = 27 tx/second

For non-segwit-enabled Bitcoin nodes, transaction capacity is determined by Bitcoin's 1MB block size limit. A 1MB block will contain about 2000 transactions on average (see here). Because 1 block is mined every 10 minutes, this works out to roughly 3.33 tx/second.

BCH transaction capacity is determined by its 8MB block size limit. Because a given non-segwit BTC transaction should be roughly equal in size to its equivalent BCH transaction, this puts the transaction capacity of BCH at 8x that of BTC without segwit, or roughly 27 tx/second.

For segwit-enabled BTC nodes, transaction capacity is limited by the 4MB block weight limit (Note that block weight ≠ block size, see here for more info). It is estimated that roughly 2x the amount segwit transactions can be fit into a BTC block as can their non-segwit equivalents (see here and here). This would make the transaction capacity of a fully segwit-enabled BTC network roughly 6.66 transactions/second.

A few things worth noting:

1) Visa typically processes ~2,000 transactions/second and can process up to 24,000 transactions/second (see here).

2) As of Feb 2018, roughly 14% of BTC transactions are segwit transactions (see here)

3) On average, a serialized BTC block full of segwit transactions should take up ~2MB on disk. 

4) Increasing the block size limit increases the amount of disk space and RAM required by miners to operate full nodes.

5) Generally speaking, the BTC community plans to increase transaction capacity via an overlay network (e.g. the lightning network) whereas the BCH community plans to increase transaction capacity by continuing to increase the block size.