To answer your question: both.
If I think that the actual map in-book has something that would lead to a better understanding of the terrain or area, I'll allow my players a glance while blocking off areas that they either cannot realistically see or have yet to explore.
In addition to that I, and every DM I've played under, use a game mat and draw the map out onto the mat with wet erase markers. While it may not be as detailed as the in-book map, that's why I supplement the drawn one with the one in-book.
Also, I know some games come with pre-printed maps that you can use as-is. Those are really nice if you can find an adventure/module with them, but I've noticed less and less adventures have them. (Probably to save on printing costs)
Now, if the map in question is a region map, I just use it as is so the players know their relative location in scope of the adventure world, but only if it is justifiable in terms of character knowledge. If it's the area around their base of operations, yes I'd use it. If it's some new area that they are exploring and have never been to, then no. But I would have a version of said map for them that highlights the locals that they have visited.
Overall, I'd use your judgment on showing the in-book maps to your players so you don't reveal too much while flavoring your (potentially poorly) hand-drawn version.
Your first stop for maps of Rifts Earth is the Rifts Game Master Guide. Pages 332 through 352 cover the borders of most notable locations in Rifts Earth: the Coalition States, Federation of Magic, Manistique, Northern Gun, Lazlo, Tolkeen, the Canadian Hivelands, the New West Territories, the Vampire Kingdoms, the Pecos Empire, Atlantis, the British Isles, Western Europe, Russia, Japan, Australia, and the Phoenix Empire in Africa.
The borders (current and hoped-for) of the Coalition States are also shown and discussed on pages 13–14 of Rifts Sourcebook.
Siege on Tolkeen: Chapter One describes on pages 112–114 the extent of the Kingdom of Tolkeen, and features a nicely detailed map of Rifts Wisconsin. Detailed information on European, Japanese, and Australian territories also appear in the last installment of this series, Siege on Tolkeen: Aftermath (pages 183, 188, and 196 for maps and surrounding descriptions).
Best Answer
It depends on how much the characters can see, how surprising the details they can't see are, and how much you trust your players to not act in-character on out-of-character information. If you want to keep surprises from them, re-drawing the map as the characters move (as you suggested) is the classic way of dealing with this. But there are other options.
You could also lay out your existing map with areas covered by strategically-placed index cards or sticky notes. This is a nice, low-fi solution and usually works well as long as there isn't a breeze.
There is also the option of scanning the map into your computer and print out individual areas. This has the advantage of being able to cope with events that will happen to change the map after the characters have already seen the area, such as rock slides and cave-ins. You could also do this with an online program such as roll20.
Finally, a time-honored tradition is to describe the area and have a player draw the map as they go. This has the advantage of keeping the players involved during exposition, and giving them a bit of agency over their adventure.