Self-inquiry is essentially a Vedantic approach which seeks to directly transcend relativity whereas the approach of Tantra goes by the scenic route.
Veda means knowledge and Vedanta means the end of knowledge and relates to the single minded pursuit of transcending relativity and Realising the Transcendental Self; the one without a second. The teachings of the Upanishads (a class of Indian spiritual texts) expound on this, but the same intent of transcending relativity can be pursued outwith this context of traditional Indian culture.
There are several schools of Vedanta, each looking at things from a slightly different angle, but the school which this teaching of self-inquiry, as expounded by Ramana Maharishi and others, belongs is that of Advaita Vedanta. Advaita means 'not-two' or 'non-duality' and refers to the viewpoint or experience where everything is percieved as the manifestation of the One Essence with no degrees of separation.
Tantra on the other hand comes from the angle of the multiplicity of life, rather than the unicity. Ultimately the aim is the same but the approach is different. I only have an incomplete and outline understanding of Tantra (as of Vedanta), but I understand it to deal with the diversity of the ways that the One Unified Source manifests in this creation.
The essential insight of Vedanta is that the Self alone is Real and all else is an illusion, so it tends to not give much attention to, or is even dismissive of, the relative aspects of life. On the other hand, from the stand point of this ultimate realisation, even this world is seen as a manifestation of the Self, and the Jnani (Enlightened Sage from this Vedantic pathway) makes no distinction between the Self at rest (Absolute Transcendental Silence) and the Self in action (creation; the world; relativity). For the Jnani it is all a play of Oneness.
Therefore, given that diversity/manifestation is as much the Self as the Absolute is, then approaching Self-Realisation from and through the apparent experience of differences is an equally valid approach. This I understand is Tantra.
In the West tantra is associated with sex, however this is only one aspect and not the main point in itself. As far as I can work out the connection with sex has two aspects. The first aspect is in encountering and releasing our inhibitions, fears and neurosis around sex which are all part of the false-ego which keeps us in the illusionary experience of separation.
The second aspect is in the actual energetic interplay between the male and female polarities. Just like electricity has two opposite poles between which the electrons flow, so both within ourselves and between the sexes somehow the Self has manifested as this polarity of male and female and tantra has evolved ways and means to harness this polarity for accelarating the process of evolution (back towards our original true nature and that much the richer for the experience).
Broadly speaking the male aspect is considered to be Silent Awareness and the female aspect is considered to be Dynamic Energy. Both of these poles are contained within each of us and on a cosmic level are said to be the two inseparable aspects of the One Divine Self.
Tantra concerns itself with all of the various manifestations of energy inherent in this human form as expressed through the various energy pathways (nadis) and centres (chakras). The main chakra system, running up the spinal column, represents the various planes of energy and perception available within the human experience.
Connecting the dots of Satchitananda in reflection as this human form, it is apparent that the energy centre connected to the navel (and the inner body space between the naval and the base of the torso) relates to the 'Sat' aspect. That is the aspect of existence or what I have been referring to in this context as the experience of 'embodied beingness', 'I amness' or simply 'amness'.
The energy centre connected with the 'third eye' between the eyebrows (and the inner space from the base of the neck to the top of the head) relates to the 'Chit' aspect. That is the aspect of awareness, or the knowing capacity.
The energy centre connected with the centre of the chest (and the inner space from the navel to the base of the throat) relates to the 'Ananda' aspect. That is the aspect of bliss, love or the capacity to feel. N.B. Bliss and love in this context don't mean overwhelming ecstacy and emotional sentimentality. Rather more likely to be a still contentment and a pure connecting energy, however, anything can and does happen.
Of course this trinity of energies are not separate from one another and are all intrinsic aspects of the one thing. They are part of a smooth circular continuum with each quality coming to the fore at times and seamlessly melding into the others. Similarily there is no clear cut physical location to them either; any or all of these energies could fill the whole body of experience. The physical locations are only approximate counterparts which can act as a guide to help one tune into particular aspects of a more subtle experience.
So tantra views this human form as a gateway to the Ultimate and honours it as such. Tantric practitioners are not in a hurry to transcend relativity but are interested in investigating and integrating all aspects of their relative experience as the very pathway to the Self.
In Tantra there is the possibility of getting lost in the diversity of life and in Vedanta there is the possibility of missing the beauty of life. Personally, and if at all possible, I favour a balanced approach which encompasses the best of both worlds.
Satchitananda - Awareness of Being is inherently Blissful
I would just add that investigating the various subtle aspects of our experience is still a different thing from self-inquiry which is directly turning the awareness around 180 degrees back towards its own source in Pure Awareness. There is not necesarly a conflict just don't try and look in two directions at once! Do one thing or the other at any one time or at least that's how it seems to me at the moment. Each one must find their own way home.
Bare bones self-inquiry
When thoughts arise inwardly ask oneself 'to whom are these thoughts arising?'
The answer naturally comes 'to me'
Then ask oneself 'who am I?'
Nip the thoughts in the bud and refocus attention directly upon the feeling-sense of 'I'
Persist in this inquiry with the intention that the True Self be revealed
Keep on persisting in this inquiry until the false ego has been entirely destroyed and the Pure Self alone remains
That is it