EDIT: excuse my ignorance, I don't have much experience using these tools.
Communication tools exist along multiple axes of effort, permanence, content-bandwidth, and discoverability.
A conversation takes little effort, is high content-bandwidth (body language, tone, whiteboarding, pointing, etc), but is impossible to discover for others and is impermanent. A formal specification is more permanent, and more discoverable, but has a bit less content-bandwidth and is a high effort activity. Old school SMSs are low-effort and permanent, but not discoverable and prone to confusion because of low content-bandwidth.
Email is kinda 'middle of the tree' all round. It's primary failing for the Entperise is the point-to-point nature of discussion (removing the ability for third parties to discover it). This can be mitigated with use of appropriate groups or tools (internal mailing lists, Sharepoint-ish blobs), but that's a challenge in and of itself...
So how is Slack better than Email? It's better in the way my cheese knife is better than my bread knife. And Slack is as much a threat to email as my cheese knife is to my bread knife... They are complimentary tools that address slightly different needs on a related spectrum, and will be painful if misapplied to the wrong problems. I can cut cheese with a bread knife, I can cut bread with a cheese knife. I can make Slack my total communication system, I can pretend that email is enough ;)
Right tool for the job, don't do dumb things, and if you can't figure out how immature use of a pling-pling-pling-pling-pling chat system is gonna wreck your day-to-day work experience... ... I assume we have worked together before and you were in charge of choosing the internal communication system...
Having said that, they aren't very natural substitutes at all. In Slack etc, the natural element of dialog is a single sentence or less. In email the natural element is a paragraph or at least a couple of sentences.
In email it is normal to select who you are sending an email to, and the recipient chooses how to organize the messages. In chat it is typical in channels organized by the sender and recipients choose to join or not.
There are many other differences too.