Are Lord Shiva and Lord Shankar two different deities?
Yes, there is a technical difference. Generally these two terms are used interchangeably as a synonym. But Shankara, Rudra, Mahadeva, etc. mean the God who dwells in the Kailasha mountain. And Shiva means His absolute formless mode of bright light.
The Linga Purana has the description regarding the origin of Siva which goes something like below:
Just before the beginning of the creation, Brahma and Vishnu had some disagreement and a quarrel ensued. At that time a column of bright light appeared before them and solved their issues. That bright light is described as below:
kṣayavṛddhivinirmuktamādimadhyāṃtavarjitam [LP - 1.17.34]
That linga was surrounded by thousands of flames and hot like fire of death. Without any beginning, middle and end, that was free from decay and growth.
This endless formless form of bright light is known as Shiva:
śivaiko brahmarupatvānniṣkalaḥ parikīrtitaḥ [Shv. Pu. - 1.5.10]
- Shiva alone, being Brahman, is known as formless and quality less.
Shankara or Rudra
The Vishnu Puran has the following description regarding origin of Rudra:
Later on Brahma commenced the process of creation. During that time he desired a child like himself. Then a blue-red colored child appeared on his lap and started crying as he wanted a name. Brahma named him Rudra, but he still continued to cry. So Brahma gave him another seven names. They are: Bhava, Sarva, Ishana, Pashupati, Bhima, Ugra and Mahadeva. Rudra married to Sati, the daughter of Daksha. Sati getting angry on Daksha gave up her life and reborn as Parvati, the daughter of Himavat and Mena. Then Hara (Shiva) again married Parvati. [VP - 1.8.2-14]
Also, in the Gita Shri Krishna says He is Shankara among all the rudras:
rudrāṇāṁ śaṅkaraś cāsmi [BG - 10.23]
- Of all the Rudras, I am Shankara
So it means Shankara is a Rudra. Hence, although Shiva and Shankara both imply the same deity, Shiva means his formless pure bright light form (meditated upon by Yogis) and Shankara means his commonly known Kailasa dwelling ascetic form (worshiped by devotees).
To remove the doubt that some may incur, I am explaining it further. Shiva, Mahadeva, Shankara, Rudra, or whatever name we may say for the Kailasa dwelling God, the fact remains the same that He has a formless aspect. So we can say Shiva, Mahadeva, Shankara, or any other name has a formless aspect. For example, Keshav cites a statement from a book which says as per Kurma Purana Mahadeva is by nature formless. The verse actually exists in the purana and it is as below:
arūpaḥ kevalaḥ svastho mahādevaḥ svabhāvataḥ [Kurm. Pu. - 1.10.82]
- Mahadeva is by nature formless, single, dwelling in his own self.
This verse doesn't say Mahadeva is the formless aspect, but it says by nature He is formless. So this means, He has a more general aspect that is endowed with form. And we all know that general form having the moon on his head. For example the following verse states the form:
tadantare mahādevaḥ śaśāṅkāṅkitaśekharaḥ
prasādābhimukho rudraḥ prādurāsīnmaheśvaraḥ [Kurm. Pu. - 2.1.31]
Then Rudra, the great lord Mahadeva, whose head is endowed with the moon, appeared with a smiling face.
But forms like these are illusory as stated in the previous line of the book or in the verse [1.10.82]. So if forms like these are illusory and dependent, then which one is the independent formless form? Well, that formless aspect, although can be referred by any name, generally referred as Shiva. So Shiva also means the formless form that Yogis meditate upon as already mentioned in the answer :
jyotiḥ paśyanti yuñjānāstasmai yogātmane namaḥ
yayā saṃtarate māyāṃ yogī saṃkṣīṇakalmaṣaḥ
prapadye tat paraṃ tattvaṃ tadrūpaṃ pārameśvaram
nityānandaṃ nirādhāraṃ niṣkalaṃ paramaṃ śivam [Kurm. Pu. - 1.10.70,72]
To Him whom the yogis see as the bright light and by whom the yogis becoming sinless cross over maya, I offer my obeisance. I surrender to that supreme essence, supreme lord, always blissful, independent, without from (parts) supreme Shiva.
Because a person's name which is a Sanskrit word can have a meaning, someone may argue that the name Shiva in the above verse means auspiciousness, a quality of the deity not the deity Himself. But if the meaning is taken literally, then the same will also apply to Mahadeva. Beacuse Madhadeva can literally mean the title or definition (saṃjñā) meaning great god or it can also mean the Rudra named Mahadeva that originated from Brahma as stated above from Vishnu Purana. The same names are also given in Kurma Purana:
bhavaḥ śarvastatheśānaḥ paśūnāṃ patireva ca
bhīmaścogro mahādevastāni nāmāni sapta vai [Kurm. Pu. - 1.10.26]
He has these seven names: Bhava, Sarva, Ishana, Pashupati, Bhima, Ugra and Mahadeva.
So literally the title Mahadeva can be addressed to the formless Brahman or any other god, but generally it means the moon crested form of Shiva. Nevertheless, to make it more clear I am citing another verse which makes the subtle difference more apparent:
mahādevaṃ arcayitvā śivasāyujyamāpnuyāt [Kurm. Pu. - 2.35.31]
- By worshipping Mahadeva, the worshipper obtains sayujya (identity) with Shiva.
We know sauyjya means a type of mukti where the soul merges in the formless Brahman, and here the term Siva is explicitly used apart from Mahadeva to refer to the formless Brahman where the jiva merges. Even to describe the one and only supreme Brahman, the word Shiva has been used:
nityānandamamṛtaṃ satyarūpaṃ śuddhaṃ vadanti puruṣaṃ sarvavedāḥ
tamomiti praṇaveneśitāraṃ dhāyāyanti vedārthaviniścitārthāḥ
na bhūmirāpo na mano na vahniḥ prāṇo'nilo gaganaṃ nota buddhiḥ
na cetano'nyat paramākāśamadhye vibhāti devaḥ śiva eva kevalaḥ [Kurm. Pu. - 2.10.15,16]
All the Vedas state that the purusha (or supreme entity) is always blissful, pure, and has the form of truth. Those who have understood the meaning of the Vedas meditate upon Him through the pranava (OM). Neither the earth, nor the water, nor the mind, nor the speech, life air, wind or sky, nor the intellect, nor the consciousness, nothing shines in the supreme sky, only one Shiva alone.
So from examples like these and above cited references, it is just apparent that there is a subtle difference between Shiva and other terms, even though they are used interchangeable at many places. But the names are used so interchangeably that it doesn't make much sense literally and depends upon the context. So the point is, even if the difference is subtle, it certainly holds true from the Yoga point of view where Shiva is most often used to denote the supreme Brahman.
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