Ah yeah, that's true. This is because most Spanish speakers pronounce /t/ as denti-alveolar (the tip of the tongue touches the back of the teeth) while most English speakers pronounce /t/ as purely alveolar (the tip of the tongue stops at the back of the alveolar ridge). I'm not sure why the IPA chart doesn't differentiate these specific phones, honestly. I imagine it's because although the sounds are very different cross-linguistically, maybe there aren't any languages where these phones are contrastive and thus they don't warrant separate glyphs in the IPA. Still, I'd love to see them!
That said, I think this is actually a relatively uncommon case in the IPA, because in addition to this simple chart the IPA also specifies a number of diacritics for more complex and precise transcriptions. These diacritics are often not used in broad transcriptions within a single language because most languages don't differentiate phones at this level in a contrastive manner. But in a narrow transcription, the diacritics are used as appropriate, generally to help the transcriber make a point about what is noteworthy in that transcription.
For instance, in English let's consider the "t" as you mention. We'll look at "tap" and "pat". The broad transcriptions for these words are /tæp/ and /pæt/, respectively. But the narrow transcriptions would be more like [tʰæp̚] and [pʰæt̚] (Note that broad transcriptions are phonemic and given between slashes, while narrow transcriptions are phonetic and given between brackets). What this shows is that, in English, syllable-initial voiceless stops are aspirated (indicated by the little "h"), while word-final voiceless stops have no audible release (indicated by the angle symbol, known as "corner"). So in addition to the main letter glyphs of IPA, there are tons of diacritics that help you write your transcriptions more precisely!
