tongue and teeth are in the same place for both, but the difference is just whether the voice box is vibrating. Voiced - the start of the, they, that, though, thus. Unvoiced - the start of thing, think, this, three, etc.
If you can't feel the difference by putting a hand on the throat where the vocal cords are, then another way to try is to block your ears with your fingers. The voiced version will buzz more than the unvoiced.
Note also that when you say the vowel, it will be voiced, so you will feel the vibrations or hear the buzz at that point regardless - so it's only when you are saying the TH part of it that you should look for the difference.
If you extend the TH for a bit of time and don't say the rest of the word, you might be able to feel it better.
The difference in voicing is the same for many pairs of phonemes in English. Mostly we have different letters/combinations of letters for these, but TH is one that we spell the same, even though they are as different in sound as s/z; f/v; sh/zh; b/p; k/g; and all the other pairs. Some languages do have different symbols for the voiced and unvoiced TH sounds as well, which makes it easier to describe them.
f/v are a different pair of letters that have that same difference in voicing. Also, they are made in a very similar way in the mouth as the two TH sounds. The difference is where the tongue and teeth are. For f/v, the teeth are touching the bottom lip; for the TH sounds, the tongue is slightly between the teeth.
People who mix up those sounds are likely to mix up the f and the unvoiced th, because they are both unvoiced. So they say free instead of three, or fin instead of thin. (Or vice versa, occasionally).
When they substitute v for th, it tends to be for the voiced Th, because v is also voiced. So you get 'muvver' for 'mother', or 'vat' for 'that'.
Thus is pronounced with the voiced TH, like there and then, and that is probably what they were (rudely) correcting.
Sometimes whether a 'th' is voiced or unvoiced depends on accent and dialect. I know places were the 'th' at the end of 'with' is unvoiced.