What a fun question! The limitless boundaries of space hold both mystery and innate intrigue, and there are so many incredible possibilities.
The advantages of space tourism center around discovering more about the unknown reaches of our universe. The thought of being able to personally navigate areas that few people have been able to ever see is the same quest that has driven the exploration of this planet for thousands of years. Humans have a desire to discover more about unknown places and to solve mysteries.
Space tourism could also satisfy these desires for a certain (wealthy) segment of the population. There is certainly an economic impact that would work to the advantage of companies working to supply the needed modes of transportation, so people with aerospace or computer engineering degrees would benefit from more career options. Universities with those programs would likely see an upswing in interested students.
Opening up privatization of space travel also allows for growth without governmental interference; currently NASA is funded only as the government allows, so when politicians steer toward a leaner economy, space funding is often one of the budget items to receive a cut. However, private funding would allow for the travel and exploration to continue regardless of whether any politician sees the benefit of it or not. If companies have the money, they are free to move forward with their own plans.
There are disadvantages, and many of those center around the risk to humans. Space is unpredictable, and the technologies needed to reach some of the goals on the horizon haven't even been fully developed yet.
There are health risks to humans in space that we already know about (radiation, risk of being hit by meteors, cosmic rays), and long term weightlessness has its risks, too. "Space tourism" also opens up a vast field of possibilities, from simply going up and experiencing five minutes of weightlessness and then returning to Earth—to going on a one-way trip to Mars for which a return is currently impossible. Where will it stop? Will people sign up for one-way trips to Jupiter, which would take approximately 6 years to reach?
NASA sent Voyager 1 and 2 to examine the far reaches of our solar system in the 1970s, and they estimate that around 2012 it entered interstellar space beyond Neptune, meaning it took around 40 years to make that journey one-way. It is worth recognizing, therefore, that the distances in space are incredible, and we currently don't have the technology to make those journeys happen quickly.
Another argument against space tourism is that it currently isn't very eco-friendly—for Earth or elsewhere. Opponents of the idea often cite evidence that we haven't done a great job in taking care of our own planet and maybe shouldn't be trusted to take those same poor values to other areas of space. The most conservative rockets in development for space tourism require around 100,000 gallons of fuel, most of which comes from non-renewable resources. So as more of those resources are being diverted to fund the desires of the wealthy in space tourism, that could mean that the rest of us will feel the economic impact of having less supply—and therefore higher prices.
There are many great questions which could be answered via further space travel and tourism opportunities, and technologies used to accomplish space exploration are changing quickly. This is a fascinating area of scientific development to consider.
Space tourism, as the name implies, is traveling to space as a vacation destination to see a new site, not as part of a scientific mission funded by the government, as the NASA moon landings were.
Other than the advantages and disadvantages already listed, space tourism offers the advantage of a "wow" factor that earth travel at this moment has a difficult time delivering, especially for people who have "seen it all." Even once-remote places like the Himalayas have become popular destinations littered with trash and sometimes crowded with visitors. There is almost no part of the earth that has not been explored. For those looking for novelty, space is incomparable. For those who thrive on thrills, the danger of space travel rivals that of any safari of years gone by.
On that note, however, amortizing the risk of space travel is a disadvantage. Insurance costs are difficult to calculate, as outlined in the article "Insuring Space Flight: The Underwriter's Dilemma" by Paul Ordyna, but they include moral hazard (the fear that companies will take greater risks or be more careless if they know insurance will cover them), adverse risk (the worry that certain risks will be understated or poorly understood), and catastrophic loss (a disaster is so bad the insurer cannot pay the losses and is forced out of business). Fortunately, governments are beginning to address the insurance problem using the Outer Space Treaty.
An advantage of space tourism is the jobs that the industry is going to provide. There are already multiple companies all competing for space tourism business. Those are all companies and jobs that didn't exist a decade ago. Pilots are needed, programmers are needed, manufacturing jobs are needed, and many many other people will be working in order to see that those companies are successful.
Another advantage of space tourism is going to be its ability to renew interest in space and space exploration. With renewed interest, financial backing will follow, and that will support further innovation in the industry and space exploration in general.
One disadvantage to space tourism is currently that it is a rich man's game. The start up costs are prohibitively high for starting up this kind of company. Not only is it an expensive business to run, but it is also an expensive ticket to buy for the consumer. Right now, Virgin Galactic's ticket cost for a single seat is $250,000, and the entire flight lasts only two and a half hours. The entire flight. You are weightless for less than 10 minutes.
Another current disadvantage is that the industry is in its infancy. There isn't much regulation happening, and the business hasn't been around long enough to have its safety protocols dialed in. It's a very dangerous industry currently, and Virgin Galactic proved that when their test platform, Space Ship 2, crashed and killed the pilot.