Account Groups provide a means for grouping accounts that should have similar configurations, and allow you to modify a single setting on all members of a group at once by modifying the group.
Account Templates provide a mechanism for saving application settings as a template for later account creation. These templates can then be applied when a new account is being created to minimize the need for manual setting configuration.
For example, if you add child accounts regularly, you may want to make sure that every time you create one they all have certain settings configured correctly as soon as they are created. You would create an Account Template to do that. When creating a new account, you would pick the appropriate template, and all settings of the new account would be set according to what is in the template.
However, if you want to be able to apply a single specific setting across many of your accounts, you do that through account groups. When you first create an account you add that account to an existing account group. If you choose to change a single setting on an account group, then every account that is a member of that group will have that setting changed to the value you set. This does not mean the setting applied via the group may not be still changed manually at the account level.
The setting value in a group is never a summary of what is on group member accounts. That is because of the fact that you can choose to allow those settings to be modified at the child account level. Remember that there are other options for configuring settings on child accounts: you may hide specific settings on children and set them to inherit their parent account's setting value.
Also, remember that changing the value on an Account Template only affects future accounts built from the template. Existing accounts may only be modified globally through Group settings.