Ed's suggestion is likely what you are looking for to most accurately replicate your previous process.
Comparing Points in Time
To compare any point in time of a branch of code, simply right click the folder (or file) in Source Control Explorer (found under any Team Project) and select compare. If you have previously gotten this version from source control, it will default to comparing the latest version on the server to the version on disk. To compare two points in time via a label, simply change the from or two to the Server Location and change the Source Version Type drop down to Label. You can then browse all of the labels to choose from.
Extracting via a Label
You can "Get Specific Version" of any file or folder using an existing label. To do this, right click the file or folder you want and select Get Specific Version. At that point change the Version Type to Label and browse for the target label. The checkboxes are for when things go loopy, but you shouldn't have to check them. Some people get nervous and do it regarless.
Branching at a Label
You can branch of file or folder (or branch) at an existing label by right clicking that item and selecting Branch and Merge -> Branch (there are other things to do under Branch & Merge that's why the submenu exists). Change the Branch from Version to Label at which point you can browse the existing labels just like comparing. Choosing a label, you can then specify where the branch should be created and have an entirely version controlled snapshot ready for active development.