Microsoft SharePoint supports two basic User Interfaces (UI) designs when it comes to team sites; The “Classic” experience, and the “Modern” experience. The modern experience is designed to be compelling, flexible, mobile, and easier to use.

The main difference between classic and modern team sites is the integration with Office 365 Groups. Modern teams sites are connected to Microsoft 365 groups

Another area where they differ is in branding. Branding a Classic site involves the use of  site themes, alternate CSS, and master pages. While these elements are not supported on modern sites, you can still brand your site using new and easier ways. As an example, you can quickly and easily customize the colors of your SharePoint site by changing the theme to reflect your professional style and brand.

With the move towards modern UI experience, the following experiences are modern by default in Classic Site:

  • The SharePoint start page in Microsoft 365
  • Most lists and libraries
  • New pages
  • Site contents
  • Site usage
  • Recycle Bin

How to move a classic team site to a modern team site

Site collection administrators can connect a classic site to a group, and update the home page to the new home page experience at the same time.

So what happens when an Office 365 group is linked to a SharePoint site? How is the Office 365 managed in relationship to the SharePoint groups? 

The section below is an extract from Microsoft doc that explains the relationship between the Office 365 and SharePoint groups, once an Office 365 group is connected to SharePoint site.

Site permissions and group membership

Site permissions and group membership are managed separately. Group members always have access to the team site, however the team site itself can be shared with additional users who are not group members.

Sites and groups have different types of permissions. The following lists show the relationship between site permissions and group membership:

Site Collection   

These are the default permissions groups (with their default permissions levels) for a site collection:

  • Team Site Owners (Full control)
  • Team Site Members (Edit)
  • Team Site Visitors (Read)

These groups support nested membership, such as Azure Active Directory groups or Microsoft 365 Security Groups.

Microsoft 365 Group   

These are the group roles (with their permission levels):

  • Group Owners (Full control)
  • Group Members (Edit)

These roles do not support nested membership, such as Azure Active Directory groups or Microsoft 365 Security Groups. When you add users to these roles, they are automatically added to the related site collection permission groups: group owners become site owners and group members become site members.

Connecting a group to an existing site    

When you connect an existing team site to a new Microsoft 365 Group, the following permission changes happen:

  • The site collection administrator that connects the team site to a new group becomes an owner of the group.
  • SharePoint Online will suggest users to be added to the Group Owners and Group Members group based on the site’s current users. If the site permissions groups contain security groups or Azure Active Directory groups, SharePoint Online is not able to suggest group members based on the contents of these groups – you must add these users to the group manually.
  • Any new group members you specify are added to the associated site collection permissions groups – Group Owners are added to the Team Site Owners group, and Group Members are added to the Team Site Members group.
    These changes are made at the site collection level. If you have subsites and you have broken permissions inheritance, your permissions model remains unchanged. Permissions filter down from the site collection level according to the model that you created, including the new group owners and members.
  • If you set the privacy level of the new group to Public, then everyone in your organization – except external users – will have Group Member access to the group.

Reference:
https://support.office.com/en-us/article/connect-a-classic-experience-sharepoint-team-site-to-a-new-microsoft-365-group-469c6ee0-2139-4496-9914-7e39d07ac49d