Is the Vex or FIRST robotics competition a better choice for a child in high school?

Both VEX and FIRST have very strong robotics competitions for the high school level.  If you are a parent choosing between the two, multiple factors need to be taken into consideration.
The first factor is the availability of a robotics program in your area.  While you can form an independent team that is not affiliated with your school or another organization, it will require a significant financial and time investment.  At the high school level teams typically design their robots with CAD software and must machine the parts themselves.  Teams often consist of 20 or more members so that all of the aspects of the competition can be covered, such as creating and assembling the hardware, electrical wiring, systems design, programming and documenting the engineering process.  All of the parts and equipment need to be purchased and maintained, and members put in a lot of hours during the competition season.  Joining a pre-existing team that already has equipment, members and mentors familiar with a program will make it easiest for a child to participate, regardless of which competition the team is a part of.
If both VEX and FIRST programs are available, then it depends on how intense of an experience the child is interested in.  The VEX competition has a lower barrier to entry because it requires smaller robots.  Two pairs of robots compete against each other in a match to see which pair can get more points.  While teams will need to machine some parts and must have skills in electrical and software design, a team can be smaller and still compete effectively.  The FIRST competition involves much larger and more complex robots and challenges.  Two alliances of three robots each compete against each other in a match, with a total of six robots on the field at once.  Teams need to have more members to effectively compete, and participants need more in-depth training on hardware, electrical, system and software design.  There is also a business requirement, where teams have to create a marketing banner, brochure and video for their team to submit as part of the competition.
Whether you choose VEX or FIRST, a child in high school that participates in a well-run robotics program will gain valuable engineering and organizational experience that will serve them well, both on a college application and after graduation.