Waxing isn't exactly a standard dinner-table topic, so most people looking for a first-time wax tend to get their information on the subject from the internet. The problem with that is, when it comes to Hollywood and Brazilian waxes, people often conflate the two. This very quickly becomes a problem when, come waxing time, you ask the technician for something you don't actually want. Yes, Hollywood and Brazilian waxes both involve the removal of most of the hair "down there," but there are, in fact, differences between the two waxes.
Namely: Brazilian waxes leave a small strip, triangle, square, rectangle, or oval-shaped piece of hair in the front, while removing all other hair (including the hair on your backside.) Conversely, when you're asking for a Hollywood wax, you're asking the technician to remove all of the hair: the front, the middle, and all the way to the backside. A Hollywood wax leaves you entirely bare. The only difference between a Brazilian and a Hollywood is the patch of hair, occasionally referred to as a "landing strip," that's left in the front with the Brazilian.
Unfortunately, many salons have started to refer to a Hollywood style wax as a "Full Brazilian", which is why the two styles are often confused for being the same wax.
It's particularly important to keep this in mind when deciding on a salon and a treatment, not solely because price points vary for different styles, although that is a big factor. Just cleaning up a little bit of hair outside the underwear is going to take way less time and product than thoroughly removing all of the pubic hair. Time is money, and nobody wants a surprise when paying their bill.
But considering the salon's specific menu has more than just fiscal implications. Just like a hair appointment, the salon wants to not only block out the right amount of time for your service so they can get to the next one, but also assign the right technician. A Hollywood wax takes more time, and possibly a different technician, than a bikini line wax does. Not every technician is willing to do every kind of wax, either due to lack of training, simple discomfort, or even gender. Conversely, some technicians might specialize in Brazilian- and Hollywood-style waxes.
It's be frustrating for both you and the tech if you show up only to learn that they aren't willing to or capable of doing the exact wax you want.
While there are plenty of names or styles for bikini waxes, most people customize their wax based on how much hair they want to be removed and where they want the hair to be left. However, keep in mind that if you're new to waxing, it's most comfortable to work your way from a regular bikini wax to a Brazilian, and then to a Hollywood. Waxing your bikini area bald from the get-go has the potential to be a little more pain and irritation than you bargained for. Thankfully, removing hair from the root gets easier over time.
Regardless, if you want to go full-throttle immediately, more power to you; you never should feel locked into any one kind of bikini wax. At the end of the day, go with your personal preference. Just be educated on what exactly that is, and remain aware that the price does go up the more hair being removed, or the finer the detail.