Much of this discussion depends on which version you look at. The T-90S, T-90MS, Arjun Mk 1 or even Mk 2 should be relevant for this discussion.
Some very undetailed rough thoughts:
Protection
Generally it is hard to measure armour protection. The T-90S has relatively modern passive armour which is based on the expertise of the Russian Nii Stali institute, which has composite armour for decades. Besides the composite armour, the T-90S employs Kontakt-5 explosive reactive armour, which is known for offering a relatively high level of protection. The T-90 also employs ERA tiles on top of the roof providing protection against artillery submunitions and single warhead top-attack missiles.
The Arjun is protected by Indian designed Kanchan composite armour. There is a huge lack of information about Kanchan and in the end everything in the internet can be traced back to one or two different sources. It is hard to say that Indian armour is better or worse than the T-90S' composite armour, but I don't have found any real reason to say "Kanchan is better". Images from the Arjun Mk 2 prototypes show reactive armour at the hull front and at the turret, but still the coverage is far below that of the T-90S and T-90MS.
Indian T-90S - so was I told here multiple times - also use Kanchan as passive armour, which reduces the possible advantages of the Arjun.
The T-90SM has increased all-around armour protection based on passive armour, ERA and slat armour. Besides that the mantlet area is better armoured (reducing the weak spots) and much newer Relikt ERA is used, which was developed to replace Kontakt-5.
The T-90S's and the T-90SM's turrets are shaped in such a manner that in frontal engagments (+/- 30° from turret center) the turret is always protected by composite armour. The current Arjun models however have a worse designed turret, where the thinly armoured turret side should is exposed to enemy fire at angles greater than ~18°.
Another important point for crew protection is the post-penetration survivability. Here the T-90S and Arjun both have unisolated stored ammunition (at least in the known cases for the Arjun), while the T-90MS has reduced the total amount of ammunition stored directly inside the vehicle and moved a small amount of it into an isolated box at the turret rear. The T-90S and T-90MS feature mulitpurpose liners which protect against neutron radation and decrease spalling.
In my opinion the T-90MS is here the clear winner (nothing special for being the most modern tank in this comparision), followed by T-90S and Arjun (possible is also a tie between T-90S and Arjun, depending on Kanchan).
Mobility
The Arjun has the lowest ground-pressure of these three tanks according to Indian claims, but it is important that future models - with probably increased weight - will loose this advantage over the T-90S and T-90MS. The top speed of the Arjun might be greater, but top-speed is always variable and not much relevant for combat.
The T-90S is lighter and thus is more useful for Indian terrain, where infrastructure is limited. The T-90 is smaller (easier to navigate) and (according to this thread) had equal mobility in desert trials.
Firepower
The Arjun's fire control system is relatively modern and has - according to Indian sources - be proven to be good, even better than that of the T-90S in Indian trials (but there are again far too many unknown variables to really use this as agrument for the Arjun). Still the T-90MS with completely new fire control system is in my opinion the better choice.
The Arjun has a rifled gun and currently only weak ammunition. The T-90S and T-90MS with their smoothbore guns and wider range of ammunition are favoured here.
