While an argument can be made for the atavism spell as written allowing the stacking of the benefits from both the spell's description and the template the spell grants, this reader has several reasons for viewing the atavism spell as, instead, trying to be helpful by providing in its description a summary of the advanced template.
First, the spell's description says that the subject "gains a +2 bonus on all rolls, including damage rolls and special ability DCs, a +4 bonus to AC and CMD, and +2 hit points per HD," while the advanced template says that a creature to which it has been applied gain a "+2 on all rolls (including damage rolls) and special ability DCs; +4 to AC and CMD; +2 hp/HD," making these two descriptions nearly identical. While not utterly absurd, essentially doubling the advanced template's benefits is a lot of oomph, even for a 4th-level spell.
Second, often when a spell provides an extra benefit the spell description makes that extra benefit obvious by using terms like also as in the spells restoration and spirit-bound blade or further as in the spells ironwood and true seeing. Such a term is absent in that sentence that could be detailing the the atavism spell's extra benefits.
Finally, the atavism spell "has no effect on animals that already have the advanced creature template," so it would be a strange spell indeed for it to forbid an already advanced creature from gaining greater bonuses because it can't be granted lesser bonuses.
Combined, while a case can be made for the contrary, this GM would rule that the spell only applies the advanced template not the advanced template and the benefits the described in the spell. This ruling also appears to jibe with this 2015 Paizo messageboard thread that describes the spell's use in a Pathfinder Society scenario.