Selenium Webdriver vs Mechanize
These are completely different tools that somewhat "cross" in the web-scraping, web automation, automated data extraction scope.
mechanize is a mature and widely-used tool for programmatic web-browsing with a lot of built-in features, like cookie handing, browser history, form submissions. The key thing to understand here is that mechanize.Browser is not a real browser, it cannot execute and understand javascript, it cannot send asynchronous requests often needed to form a web page.
This is where selenium comes into play - it is a browser automation tool which is also widely used in web-scraping. selenium usually becomes a "fall-back" tool - when someone cannot web-scrape a site with mechanize or RoboBrowser or MechanicalSoup (note - another alternatives) because of, for instance, it's javascript "heaviness", the choice is usually selenium. With selenium you can also go headless, automating PhantomJS browser, or having a virtual display. As a commonly mentioned drawback, performance is often mentioned - with selenium you are working with a target site as a real user in a web browser, which is loading additional files needed to form a page, making XHR requests, rendering etc.
And this itself does not mean you should use selenium everywhere - choose the tool wisely, choose it because it fits the problem better, not because you are more familiar with an instrument.
Also note that you should, first, consider using an API (if provided by the target website) instead of going down to web-scraping. And, if it comes to it, be a good web-scraping citizen: