I used to teach GATE and I was in GATE when I was in middle school. Programs really vary greatly between school districts.
When I was in GATE in middle school it meant one special class a day. We did some projects, but to be honest I remember very little about it. It was more of a "fun" class. There really wasn't a curriculum that was followed. It was just more what the teacher thought would be fun to do. I was also in the "top" section of three which basically just meant we moved a little quicker through the curriculum than the "low" group.
Our district has a highly gifted school-within-a-school for middle school students. It is very high caliber. Students must qualify through test scores or a very specific and formal portfolio process. There are a limited number of spaces and qualifiers are entered into a lottery system to see who is accepted. It has a high caliber curriculum that partially uses curriculum specifically developed for gifted students. It is more project based and requires a lot of higher level thinking skills. The teachers have had specific training in teaching gifted students.
Our other middles schools have programming for gifted students as well. Involvement is dependent on meeting certain qualifiers through a formal process. Some of the programming happens within the classroom. Classroom teachers get help from a gifted specialist to design assignments that require more than the average curriculum, but students aren't in a special class. They also have some honors classes (that are not high school level classes--students cannot earn high school credits). There is an accelerated math program that also provides an opportunity for students to go to the state university for math classes (they do not earn credit for it). There are some extra-curricular opportunities that are usually open to anyone--chess club, Destination Imagination, History Day, MathCounts, Math Masters, National Geographic Bee, spelling bee, science fairs, etc. There are also some literature circles.
I've taught in districts were the GATE program is nothing more than extra projects for whoever wants to do them.
The type of programming available will really be a big factor in determining if it will be a good fit or not for your child. The highly gifted program that our district has requires students to be very academic minded and very focused on school. I know some parents (and some kids themselves) who have chosen not to enroll their students in the program because they don't want their kids to have that much pressure on them and some who don't want their kids to be "nerds" and "miss out" on the social or athletic aspects of middle school. However, some of my former students who attended the highly gifted program were also very successful in athletic programs as well.
The extra curricular types of activities are usually a little more laid back. Chess Club is more about learning the game and strategizing. History Day and science fairs usually takes more dedication and focus, but usually for a short time span.
Honors type classes fall somewhere between. There may be more homework, or more challenging homework. In our district those classes tend to move through the required curriculum at a much faster pace with more complex enrichment type of assignments. It requires that students be pretty serious students who are pretty dedicated to their school work.
Benefits: more challenging school, less "boredom" if classes are too easy; strengthen higher order thinking skills; build self-confidence; opportunities to explore new areas of academia; more interaction with like minded students; more responsibility for taking ownership of your own education. Some of the benefits could be drawbacks is your daughter doesn't handle pressure well or if she isn't a really dedicated student.
Drawbacks: potentially more homework; being required to do more; more pressure to do well academically; maybe some distance developing between friends who are not involved in GATE programming and don't need to study as much. I've also seen some GATE students develop a perceived sense of entitlement because they are "smarter" than everyone else. But those students seem to be the same ones who have parents who think it is a life or death situation if their kid isn't identified as gifted and see identification as gifted as being a bragging point. Many of the drawbacks might not be drawbacks if your daughter is a very driven student.
Ask lots of questions about what qualifications there are to be in GATE classes. Does a student have to qualify to be involved? What are the qualifiers? What does the GATE program look like? What is required of students who are involved? What can you expect from the program? Is there a parent group that can answer questions for you? Our district has a parent group specifically for parents of kids in the GATE program. Can your daughter visit a GATE class and see if it would be a good fit for her?
Probably most important, be sure that your daughter wants to be involved in the GATE program. Don't force it on her. I had students who were just miserable in GATE classes because they really didn't have the drive or ambition to do the required work. They were only there because their parents wanted them to be there. I think GATE programming in our district is really good. I'm hoping that my kids will both qualify for it, but I also know it isn't a big deal if they don't.