Namaste,
I sometimes get confused about this topic, the difference between the jiva and the atma.
I think the difficulty is usually because of the approach. 'How' one obtains jñāna is important. Broadly, this can be thru' pratyakṣa (~ direct/immediate perception), anumāna (~ inference), and/or śabda (~ testimony). For those who accept the authoritativeness of śruti-s and smṛti-s as śabda pramāṇa esp. on topics that are beyond the grasp of pratyakṣa, all of them without exception disclose the nature, characteristics, and svarūpa-bhéda (~ inalienable bases for difference) b/w jīva and ātmā. One must also understand that the terms jīva and ātmā in the primary connotation convey the supreme brahman. Being a dependent principle, jīva-s too are referred to as such*.
Ex. of śruti positing two eternal entities? dvā suparṇa …
Ex. of śruti positing multiple dependent conscious beings? cetanaścetanānāṃ …
For instances on anumāna w.r.t. this one can refer the brahmasūtra each of which uphold difference for ex., ōṃ athāto brahmajijñāsā (~ since the sūtra declares the jīva as the knower (~ seeker) and brahman as the known, bheda (~ difference) is established. ōṃ ānandamayo§bhyāsāt (~ here even śrī saṅkara establishes that it is only the paramātmā who is ānandamaya and not the jīva), similarly ōm netaro§nupapattéḥ, ōṃ bhédavyapadéśācca, and so on …
In terms of pratyakṣa, no one can claim to be beyond the experience of mundane pain and pleasure, to which brahman is never subjected, hence difference is obvious.
Some of the bases for eternal difference b/w ātmā and jīva include (among the infinite):
svatantra ‡ a-svatantra
vibhu ‡ aṇu
biṁba ‡ pratibiṁba
nityānanda ‡ duḥkhaspṛṣṭa etc
Some similarities include:
ātmāsvarūpa: brahman is the inner controller of all that is, hence referred to as ekamévādvitīya. Jīva as a dependent being derives one's control over svarūpendriyas.
sacchidānandasvarūpa: jīva-s derive their sacchidānanda-svarūpa being dependent on brahman.
Can the jiva be destroyed?
nityatva: both (among others) are eternal principles.
I have read that the jiva is the subtle body, the mind and the senses, does the atma live within the jiva?
And are there other subtle layers of the body?
Subtle body you are referring to is the sūkṣma-śarīra which is not eternal. The svarūpa-déha of jiva-s are enveloped by three beginning-less bodies viz., liṅga-déha (~ causal body) made up of three guṇa-s, kāma (~ desires), and karmabīja (~ seeds of karma), sūkṣma-śarīra (~ subtle body) made up of sūkṣma-indriya-s, vāsanā-s, and sancitādi-karma lépa, and finally the sthūla-déha (~ gross body).
The ātmā dwells within the jīva (aṇoraṇīyaḥ) as the antaryāmin. He also dwells as the antaryāmin in each of the three dehas as well as inner controller and adhiṣṭhāna of the entire viśva in the viśvarūpa (mahatomahīyaḥ).
*Since saṁskṛtaṃ is essentially an etymological language and functions as such without exception, especially in śāstra-grantha-s it would be naïve to understand them in any other way.
श्रीमध्वेशकृष्णार्पणमस्तु ।