Office 365 Groups are most likely the next generation of Distribution Groups we typically have seen in Exchange in the past. There is no doubt that Office 365 Groups provide significantly more benefits over Distribution Groups... which is really just a list of email addresses to make sending email to several recipients easier.
Microsoft has gone so far as to state their recommended practice is to go ahead and convert all your Distribution Groups to Office 365 Groups. Sure, all Distribution Groups can be migrated to Office 365 Groups, but first let's ask "should all Distribution Groups be migrated to Office 365 Groups?"
Distribution Groups vs Office 365 Groups
Let's take a quick look at a comparison between Distribution Groups and Office 365 Groups. I've highlighted the major differences below.
It's clear that Distribution Groups are quite limited in comparison to Office 365 Groups, but this also highlights one of the major concerns with Office 365 Groups: Self-Serviceable.
If you are responsible for administration or management of your Office 365 tenant or have had to manage Exchange Server in the past, you probably are cringing at the concept of self-service for Office 365 Group creation. Membership self-service is not so much a problem as we can close off membership within the Office 365 Group configuration. But by default, allowing ALL users to create Office 365 Groups as they please is not what you should be allowing, unless you enjoy uncontrollable group sprawl.
It is highly recommended (but in my personal opinion MANDATORY) that Office 365 Group creation be limited to only a subset of users. For example, department managers or project managers only as these roles typically are the starting point for any collaborative initiative. To do so, you can follow the instructions provided by Microsoft for Manage Office 365 Group Creation.
To Convert or Not To Convert?
Going back to the initial direction by Microsoft to go ahead and convert all your Distribution Groups to Office 365 Groups... don't. At least not yet.
Before you just blindly convert all your Distribution Groups, understand that every Office 365 Group is automatically provisioned the following:
SharePoint Online Team Site
OneDrive Document Library (SharePoint Online)
Exchange Online Shared Mailbox
Exchange Online Distribution List (that's right)
OneNote Notebook
Yammer Group
Do you really want all of these resources being provisioned for each Distribution Group in your environment? Chances are, you don't (or you shouldn't).
I have seen organizations with thousands upon thousands of Distribution Groups. These groups were created not with the intent of a collaborative team working on a shared initiative, but just to simplify addressing email messages.
For example, in an Exchange Online migration project I recently completed for a multinational retail company, they had created a Distribution Group for each one of their 600+ store locations that contained the managers and assistant managers of those stores. This made sense as store employees changed, and rather than people in finance or operations having to look up the current managers and assistant managers email addresses, they could simply send an email to the store Distribution Group. But, these Distribution Groups were not created because each store management team needed to collaborate on a shared initiative, so in this case converting all these store Distribution Groups to Office 365 Groups adds unnecessary complexity and provisioned resources that will never be used.
Therefore, before just blindly converting all your Distribution Groups to Office 365 Groups, audit the Distribution Groups in your environment first. Office 365 Groups make sense when there is a need for collaboration outside of just simple email. Determine the purpose of your Distribution Groups first as in the long-run, this will save you time and plenty of headaches.
Migrating Distribution Groups to Office 365 Groups
After you determine what Distribution Groups should be Office 365 Groups, then you can go ahead and migrate. Hopefully, after auditing all your Distribution Groups, you have been able to reduce how many groups you have by eliminating stale or unused groups.
To perform the actual migration, I am not going to reinvent the wheel, but simply point you to the Microsoft guidance on performing this procedure. The process is always changing, but as of this writing, the migration can be performed either via Office 365 directly (if your Distribution Groups are homed in Exchange Online) or via PowerShell scripts.