TLDR: we only write contrastive sounds (phonemes) in /slashes/. For instance, [t] and [k] contrast in English as in /tæp/ and /kæp/. [p] and [pʰ] don't contrast in English so we don't write them in slashes. Dictionaries use phonemic transcriptions, so they don't write pʰ. However, we write pʰ and p in slashes for languages that contrast them (like Chinese and Icelandic).
So /slashes/ are used to enclose phonemes. A phoneme is a 'meaningful unit' in that it distinguishes one word from another, for example, tap is distinguished from cap by a single phoneme /t/, so /t/ and /k/ are different phonemes because tap and cap are completely different words having different meanings and we would enclose t and k in slashes becuase they contrast in English. There's a language called Samoan where [t] and [k] don't distinguish words so both [t] and [k] are the allophones ('realisations') of the same phoneme in Samoan.
Now there are other languages that have phonemic aspiration (i.e. aspiration can distinguish words) such as Icelandic and Chinese. In both those languages, [p] and [pʰ] are contrastive. For example, [pʰaːr̥] in Icelandic means 'pair' and [paːr̥] means 'bar', so their phonemic transcriptions would be /pʰaːr̥/ and /paːr̥/ respectively. However, they would mean the same thing in English because English has no phonemic aspiration, so we wouldn't write [p] and [pʰ] in slashes for English. Native speakers think of both the aspirated and unaspirated p as the same sound (the same phoneme /p/).
Likewise, [tʰ], [t], [ɾ], [t̚] and [ʔ] occur as allophones of the phoneme /t/ in English. Although native speakers do (and can) hear the difference between all those sounds, they think of them as the same sound—/t/.