Are ions actually charged particles?
Yes, ions are actually charged particles and neutral atoms are not. The TA's version is correct: the positive and negative charges cancel each other out. In a positive ion, we have one extra proton that gives the whole thing a net positive charge. In a neutral atom, electrons and protons cancel each other out exactly. How can they not to? After all, their electric charges are precisely identical, just with opposite signs.

Now when we speak about cancelling out like that, we only mean the net charge. Take a mole of neutral atoms and a mole of positive ions (just theoretically, of course, otherwise the consequences might be catastrophic), and even a crude electroscope will tell you the difference between the two right away.

If you specifically intend to ask about one ion, let's look at it from a different perspective. A neutral atom or molecule is, well, neutral, but that doesn't mean it would not participate in electrostatic interactions: there are those small positive and negative charges inside it. They seem to be moving, or at least not being all in one place, and you can feel tiny electric fields resulting from that. The corresponding electric potential decays with distance as $\frac1{r^2}$ for a dipole, and even faster than that for a quadrupole or other configurations of charges. The same applies to an ion, but it also has a net charge, and its potential behaves like $\frac1r$. When looking from a far enough distance, you will only feel the pull of the net charge. Look at a neutral atom from the same distance, and you will feel nothing.

Sure enough, if you look very closely, or even inside an atom (or an ion), you will see a lot of things going on around you, and the net charge will not be all that important.

So it goes.