Interestingly enough, IPA actually does have a way of distinguishing these sounds. The Spanish t is always unaspirated [t]. But the English t can sometimes be unaspirated [t] as in "stamp" or aspirated [tʰ] as in "too"
That is not what "phoneme" means. They are different sounds, but they aren't different phonemes in either language; to be different phonemes, the same language would have to consider them different. Neither does -- in both cases, one is "/t/" and the other is "weird /t/".
However, since there is no alveolar [t] in Spanish, it will generally just be transcribed as /t/ rather than having to put the diacritic on all the time.
What I was talking about was how the Spanish /t/ has the tip of the tongue just a bit further forward, so that it touches the back of your teeth. English /t/ is, for most people, purely alveolar, so there's no teeth contact.
It’s even more crucial with d. An English native speaker who is a beginner in Spanish will pronounce tened in a way that will be interpreted by Spanish speakers as tener.