‘Symmetry’ is a rather generalised concept. It isn’t so much the existence of symmetry in a molecule, but much more which type of symmetry we’re encountering.
From geometry classes, you should remember that there are planes and centres of symmetry. Alongside these, rotational axes, the identity and more exotic things like improper rotation are considered elements of symmetry in group theory — in a nutshell, the part of extended chemistry that discusses symmetry and its effects. From knowing some elements of symmetry in a molecule you can deduce others, but that’s not too relevant here.
The important point is the effect that the symmetry has on your molecule. Anything that can be represented as a vector — most notably the charge separation in a dipole — needs to transform onto itself, no matter which element of symmetry you apply.
In the case of $\ce{CF4}$, the molecule not only contains a number of planes of symmetry (not two or four, but six due to the picture bon showed). These planes also generate a centre of symmetry. And the only vector that can be transformed onto itself by a centre of symmetry is the null vector, which in turn means that a dipole moment can only be zero — hence a non-polar molecule.
In $\ce{CHF3}$, we suddenly encounter the reduction of symmetry, that is already evident in your flat drawings, but only truely discussable with the 3D-model. In a nutshell, you lose three of the six planes of symmetry and also the centre of symmetry they created. The three remaining planes of symmetry share a common axis — the one which also contains the $\ce{C-H}$-bond. And a vector, that points in the same or the opposite direction as this axis will be transformed onto itself. So you have sufficient reduction in symmetry to allow a dipole momentum, creating a polar molecule.
Simplified, this means that any and all molecules that contain more than one type of atom and do not contain a centre of symmetry are polar (to some extent). However, often — especially in the case of hydrocarbons with a difference in electronegativity of only $0.4$ — the polarity is so weak that the molecules can be considered non-polar.