High achievers are not necessarily gifted, although some high achievers are also gifted. High achievers are often externally motivated by the desire to get good grades or even high praise. They can also often be motivated by stickers with smiley faces.
However, high achievement is not a sign of giftedness. In fact, some gifted children are underachievers. They may be internally motivated, so unless they are interested in the task or the material to be learned, they may not do well on assignments and may not even complete the assignments.
High achievers may need an educational environment beyond what is offered in the average classroom, but that is not necessarily the same environment required by gifted kids to be successful.