Let's list out the differences.

1. Every Office 365 Group has a shared mailbox with a searchable history of email conversations within the group, so new members have access to all of the content and context that predates their membership

  • With a Distribution Group, new members only see discussions starting from when they joined

2. By default, Office 365 Groups are ‘public,’ i.e., discoverable for users within your Office 365 tenant. That makes it easy for people in your organization to search for a topic name and/or description and join any related groups.

  • Distribution Groups are often searchable by name only

3. Office 365 Groups provide a convenient self-service option for an organization’s users to create new groups, as well as join or depart groups

  • Creating Distribution Groups, as well as adding and removing members, is usually done by the organization’s Exchange administrators on behalf of the users.

4. Group members also have access to a shared calendar, document library, OneNote notebook, etc. for collaboration needs beyond email

  • Distribution Groups are designed for collaborating via email. Separate sets of manual steps are required to set up and use addition collaboration tools such as a shared calendar or a OneNote notebook.

It's time to move on to new Office 365 Groups

Links for Admins:
How to migrate distribution lists to Office 365 Groups 
Get the statistics about the groups in your tenant