Sniffing external networks from a public network (place)

There is something that I can not understand. If you are preparing, building up an attack from a public place (coffee shop, restaurant ect.). How can you be tracked by law enforcement? I'm asking this because the public IP address will be the one from the public place (sub-network) from where you are performing the attack. Once you are done with the attack. The law enforcement can only find out that the attack came from the public place(after gathering the required info.). After you come to your home get another IP address by the DHCP server which has no correlation to the public place network IP you were.

My question is, how would the attacked person get to you (reveal your identity)?