Symmetry boundaries are used to reduce the extent of your computational model, they are used when the physical geometry, and the expected pattern of the flow/thermal solution, have mirror symmetry.

They can also be used to model zero-shear slip walls in viscous flows.

FLUENT assumes a zero flux of all quantities across a symmetry boundary (the normal velocity component at the symmetry plane is thus zero).

Periodic boundary conditions are used when the physical geometry of interest and the expected pattern of the flow/thermal solution have a periodically repeating nature.

FLUENT treats the flow at a periodic boundary as though the opposing periodic plane is a direct neighbor to the cells adjacent to the first periodic boundary.

When calculating the flow through the periodic boundary adjacent to a fluid cell, the flow conditions at the fluid cell adjacent to the opposite periodic plane are used.

The second one is more used in pumps, propellers while the first one is used in pipe flow or ship's and it will act as an infinite domain.

Symmetry boundaries are used to reduce the extent of your computational model, they are used when the physical geometry, and the expected pattern of the flow/thermal solution, have mirror symmetry.

They can also be used to model zero-shear slip walls in viscous flows.

FLUENT assumes a zero flux of all quantities across a symmetry boundary (the normal velocity component at the symmetry plane is thus zero).

Periodic boundary conditions are used when the physical geometry of interest and the expected pattern of the flow/thermal solution have a periodically repeating nature.

FLUENT treats the flow at a periodic boundary as though the opposing periodic plane is a direct neighbor to the cells adjacent to the first periodic boundary.

When calculating the flow through the periodic boundary adjacent to a fluid cell, the flow conditions at the fluid cell adjacent to the opposite periodic plane are used.

The second one is more used in pumps, propellers while the first one is used in pipe flow or ship's and it will act as an infinite domain.

They are not the same.

E.g. there cannot be flow across a symmetry boundary, because the flow towards the boundary from one side must be accompanied by a mirrored flow from the other side gicing a net flow of zero perpendicular to the boundary.

For a periodic boundary, however, there can be flow across the boundary, but this flow must be accompanied by an identical flow across the other periodic boundary in your system (there has to be pairs of periodic boundaries). E.g. if a pipe segment has periodic boundaries at the start and end, the flow across the end boundery of the pipe must be identical to the flow across the start boundary of the pipe (this is a way to model long pipes without having to resolve the entire length).

They are not equal. For more details, please go through the Fluent user manual.

periodic BC means that fluent matches between the two boundaries, i.e if you have T=100deg at the inlet and u run the simulation T at outlet for example will be 150deg so fluent matches these 2 and inlet T becomes 150deg so you cannot use it if you want to monitor something like pressure or temperature because it will keep increasing.

for 3D problem when set periodic BC in all directions then the flow becomes fully developped.

its mostly used for turbulence modelling and you should be careful where to use it because it will definitly affect your results.

 Not same both are different.

In Symmetric boundary condition, all the variables have same value and gradients at the same distance from the boundary.  The function of such a boundary is that of a mirror that can reflect all the fluctuations generated by the simulation region The conditions at symmetric boundary are no flow across boundary, no scalar flux across boundary. On the other hand  Periodic or Cyclic boundary condition arises from a different type of symmetry in a problem. If a component has a repeated pattern in flow distribution more than twice, thus violating the mirror image requirements required for symmetric boundary condition. So...they're not the same!

Consider the case of the angular coordinate in a cylindrical coordinate system and a computational domain extending over a certain range of angular coordinate (wedge). Symmetry boundary condition can be imposed separately on each plane of constant angle. Then the velocity in angular direction will vanish. Periodic boundary condition can be imposed for the pair of planes. The angular velocity on both planes then is equal and if non-zero there is a "swirling" motion. However the angular velocity component can still be set to zero. Even then the two cases need not be identical because there can be a difference in another boundary condition (e.g. higher moments in a turbulence model). To answer whether there is anything strange in the situation of the original question we have to know more about the case.

Symmetry is different from periodic boundary conditions. Symmetry is a type of what we call Neumann boundary conditions in which the normal gradient of the variable at the symmetry line or plan is defined. Periodic flow is as flow over a group of ribs, or fully developed flow through a wavy channel that means the conditions at the domain exit is similar to that at the domain inlet, meaning that the flow exactly repeats itself at a certain pitch. For Fluent, you may look at the grid generation and refinement for both cases.

Symmetry boundaries used to reduce computational effort in problem and flow field and geometry must be symmetric: 1) Zero normal velocity at symmetry plane 2) Zero normal gradients of all variables at symmetry plane.

Periodic boundaries used when physical geometry of interest and expected flow pattern and the thermal solution are of a periodically repeating nature. Periodic boundaries reduces computational effort in problem.

I don't see where I can add comments, but what variable(s) are you applying these BCs to?

Well, it seems that, based on your question, you understand the difference between the two BCs, so I won't repeat what others have said about the differences. I did find a PDF about FLUENT BCs:

http://lyle.smu.edu/me/7337/fluent_bcs.pdf

And it does seem to say that for symmetric BCs "Flow field and geometry must be symmetric".

To further note, I would guess that you may be seeing similar results between these BCs if the problem (and or variables) can be described by both periodic and symmetric BCs. A simple example of this: consider a fully developed 2D pressure-driven channel flow with one axial cell. Implementation of periodic and symmetric BCs will have the same outcome (no variation along the axial direction).  As others have said, this of course doesn't mean that the BCs will always produce the same result (like in a full 3D model) but, it sounds like, the problem you're considering can be described by both.

As a final point, I would say that a better question you might want to ask is: "Which BC more accurately depicts the physical setup of your interest?".

UPDATE:

The link that I provided is now broken. This new one: http://www.afs.enea.it/fluent/Public/Fluent-Doc/PDF/chp06.pdf

Says "Symmetry boundary conditions are used when the physical geometry of interest, and the expected pattern of the flow/thermal solution, have mirror symmetry.", which is still in agreement with my original answer.

Periodic boundary conditions are used when the physical geometry of interest and the expected flow pattern have a periodically repeating nature. This means that the flows across two opposite planes in your computational model are identical. Periodic boundary conditions can be applied to a pair of boundary sections, which are referred to as the inlet and outlet of the periodic condition. The velocity profile for the inlet and outlet will be the same, but a difference of normal force or pressure will be allowed between them.

The symmetry condition is equivalent to imposing zero values for the normal velocity and tangential force.