answering a dead question with a boring answer, yay?
War between such super computers would potentially be pretty boring by your requirements, or interesting and destroy the entire internet before it got boring again; because sometimes Defense really is the best Defense.
Hacking isn't magic that everyone implies. Hacking isn't about being smart or good enough you can make a computer do what you want, Hacking is about finding out the machine you hacked was stupid enough to make it possible for you to do something everyone knows you shouldn't; it's far more about exploiting the defenders mistakes then anything else, and if your defender doesn't make mistakes then you CAN NOT hack them, end of story. Sure, sometimes you have to be pretty smart to figure out how to take advantage of the opening the defender made, but you first need an opening.
In terms of 'hacking' by a huge majority the most common now of days is social engineering, tricking dumb gullible people to do something dumb and gullible; because it's much easier to trick a person then a computer. This isn't an options by your own definitions.
The second most common is finding a computer that someone did something stupid on, like forget to change the basic password you use to connect to it, or give you more permission then they should have to modify that sudoers file. Closely related to this is finding out that the defender is using a program with a known bug and exploiting the alerady known bug because the defender forgot to update it.
This avenue isn't viable if we assume that both computers are near equal in capabilities. To be able to detect your enemies weaknesses you must already know how to detect such weaknesses, and as such you would have already found and patched them on your own system. It's much easier to scan your own computers, where you have full permission, then to scan your enemies so it's only reasonable to assume you would be able to defend yourself as well or better then find your enemies weaknesses.
More to the point the list of "dumb stuff people do" is well..known. Humans make mistakes, hackers can exploit a stupid or lazy human who does it, but computers aren't lazy or prone to random moments of idiotic mistakes. Any computer sophisticated enough to develop it's own changing attack vectors for attacking it's enemy will have reached a level of sophistication where patching dumb mistakes and wholes in their own system is trivial. Thus no dumb mistakes will be made.
The final form of hacking would be 0 day exploits, finding a legitimate new defect, previously never known, that can be exploited. In theory computers could do this, though again it would be much easier to scan for flaws in your code then to detect them in your enemy thus we would expect each computer to defend against 0 day exploits in their own code as good or better then they would find it in others. The expected defense of both computer is higher then their offense on average.
It's like giving two people pocket knives to fight with, and then armoring them with full plate mail. A pocket knife can be lethal if used very well sure, but if both sides have far better armor then they have weapons it's going to be a very long fight.
Of course this isn't failproof, it's possible one super computer figures out a 0 day exploit before the other one, a brand new bug to exploit before the other side figured out it existed to patch it; just like our armored knife fighters may eventually find a weak joint or eye slit that they can stab through after a long enough battle.
However, the fight will be anticlimatic in this case. Whichever side finds the 0 day exploit uses it to kill the other computer, end of story. No amazing battles, explosions, or other fun things. Once an opening is found you can use it to get your rot access and delete the other computer's hard drive and your done. It's unlikely that either computer will find some bug they can exploit to hinder the other computer without entirely taking it out, any minor bug that doesn't grant full root access will likely lead to the defending computer detecting the intrusion and fixing the bug too quickly to be useful, it has to be an all or nothing surgical strike.
Thus what your see, if you somehow watched the network traffic, is two heavily armored computers thinking for long lengths of time, with the occasional query or message being sent (from an appropriately subverted node so as not to be so obvious) to see rather or not the enemy computer has fixed a bug you think you may have figured out how to exploit. The digital equivelent of two fighters feinting at each other for most of a match looking for an oppening until one finally sees one and goes in for a quick kill with their knife.
Of course, this all could happen after they have destroyed much of our technology first...
Ignoring the difficulty of 'hacking' there are two other options for attacking a computer which are a bit more brute force. The first is a Distributed Denial of Service attack, as previously mentioned in other's answers. You can overwhelm your enemy computer with so many requests that they are flooded and unable to respond to do anything. Notice I said DDOS not just DoS, a DoS that tricks your enemy into using far more resources then you use would fall under the ground of 0 day exploit and most likely the computers will be defending from that from the start.
The problem with a DDOS is that it takes far more computers working together to take down one enemy computer. It's easy to detect a Dos and blacklist the attempter, so again defense is easier then offense. Your attacker DDOS options fall under the ground of either temporarily disabling causing him exactly as many resources as it costs you (making it pointless to do) or preminantly disabling you by massive overkill that overwhelms you.
DDOS strategies could be employed, but to do them both computers would ahve to rush out to grab as many nodes as possible to use them to attack the other's nodes. Since both computers needs more energy/resources to take down an enemy node the only option is for both to try to own more nodes then the other, meaning a race to take every network node out there.
However, again, because it's easier to defend then attack this will be pretty futile. Assuming anywhere close to being evenly matched and the computers will take over unaligned computers at close to the same weight, meaning for every one enemy node taken down they have secured 2 or more nodes of their own; meaning even with an all out offensive both sides will be gaining nodes faster then loosing them until they have taken over (or destroyed) every easily taken over unaligned computer. At which point we revert back to the first example of both computers trying to find 0 day hacks, except that they will run the 0 day hack on every enemy node at the same time (since they have about the same number of nodes they should be able to do this), taking over most of the enemy computer at once and then having the power to DDOS the rest.
In reality what would instead happen is that one of the computers will write the "take over enemy node" code a few dozen milliseconds sooner, which will let that computer take over many dozens of computers first (once you write the exploit you can use it simultaneously on many machines, the trick is not in processing power or bandwidth, but coming up with a smart hack). Whichever is the first to start taking over nodes will grow exponentially faster an soon have an easy time overwhelming the other.
Of course, why take over the internet internet when you just need one unalligned node. Both computers communicate with the internet through their modem...which is connected via a wire to one tiny service prodivder, who would in turn connect you with others. If you can take out that first provider, the single hope from local machine to tcp provider, you can isolate your enemy easily. Thus the real trick would be to hack whatever that first service provider hope is.
These service providers will be far more vulnerable then the enemy super computer, because they aren't being defended by a super intellegent hacking computer. Being bult by humans they will have idiotic human mistakes in them. A computer able to do any of the more advanced hacking should have an easy time scanning for one of this exploits. It's just a matter of which super computer finds a whole in the flawed human-made firewall required to get into the service provider, at which point they will ensure that provider ignores the other computers messages and that computer will be sitting in an isolated corner without the ability to do much of anything.
So basically, either one computer or the other first finds a way to hack unalligned nodes sooner, at which case they can use that to win a quick victory by having access to far more nodes, or if they don't they will gobble up all of the internet, then sit and stare at each other forever as they realize that it's just to easy to defend.
Of course the real answer is that they would get humans involved, because social engineering is far less difficult then finding a 0 day exploit for a strong AI, and if these are only weak AI then the real answer is which ever weak AI has the better programmer will win because it's really a matter of which programmer was the better hacker, just with their weak AI as a layer of indirection.