Check Airman
I find it curious that so many posters have mentioned briefing the taxi route. At my company, very few people brief the expected taxi route. I'm not one of them. Unless you're addressing a specific threat, it really doesn't add much value. What happens when you've briefed A B C, and you're assigned X Y Z?
Including the expected taxi route in your pre-starting/close-out brief while there are no other distractions/duties/cockpit chores establishes orientation as well as forces at least a basic familiarization of taxiway(s) to expect, the need to cross (or not) cross a runaway(s), etc. It's a ground floor step in keeping situational awareness once you're moving when there are distractions, and puts both pilots in the loop to begin with. There are plenty of pilots out there that aren't going to the same 10 airports for the month, or year, or decade. Many times at least one person in the cockpit is unfamiliar with the "usual" operations, this familiarizes them beforehand. In your company if you don't brief, when you get a new pilot on line, how do you even know he/she has any idea as to where they are or going, even in a general way?
Doing so also places an emphasis of the importance of the taxi phase from the start. It heightens awareness and is a reminder to remain vigilant. More incidents/accidents occur during this phase than on missed approaches, and yet you probably brief the missed approach religiously. Why brief the missed approach even though the chart and text is right there in front of you. You do it because it familiarizes and emphasizes the notion/reminder that "Every approach includes the missed approach. Landing is a bonus".
What do you do if/when your taxi clearance is changed? Well, what do you do if you get an amended departure procedure as you taxi out that's different from the clearance you relieved at the ramp? What do you do if they change approach procedures after you've briefed a different one? If you're arriving and you already have an awareness they may change your approach and you've talked about it, there's usually no confusion or drama involved at all. Taxiing is no different.
Related to the above, if it has changed and you're familiar with the alternatives, it will alert you more quickly to a completely bogus clearance. Controllers have been known to mix up Flight/tail #s with aircraft. If you're given a taxi clearance that has a route nowhere near where you are, it won't take a lot of figuring out it was meant for another aircraft somewhere on the field. No biggie if nobody's moving, but if they are underway and need the clearance you erroneously received quickly, sorting out that confusion in a timely manner may save someone some big grief.
This (establishing and maintaining SA during taxi) was talked about and emphasized during the big push years ago to reduce runway incursions. It was found that often the taxi phase was given short shrift in thought and deed just because our wheels are on the ground. Even if it isn't a phase of flight, it's still at least 2 phases of our operation for every sector and airmanship still applies. Hopefully these lessons haven't been forgotten so quickly.