The most common reason for users visiting a web archive is to retrieve a website or a web page which is no longer accessible online, or has been changed significantly.
This is a useful feature which may be very helpful, and similar to visiting a library in order to access a copy of rare or out of print material.
Another way to use web archives is as “big data”. Counts of hits for specific subjects, measurements of activity in relation to events, tracing activity patterns for specific subjects are obvious perspectives for use, and for research. (For example: “Where do debates on a subject occur; where do they link to, do they move from certain locations to other locations, etc.”).
And such use can be fully or partially automated in ways that would not be possible with data stored in an analogous form.
A combination of big data analyses and closer studies of selected or sampled examples will often be a good strategy for research purposes, combining solid observations of general trends with closer in-depth studies.
See also the page Ideas for examples of research possibilities using archived web.