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The Enterprise-D has a computer core with a capacity measured in "quads" - this is for storage capacity, and processing speed. (This was done intentionally to remove any references to real-life tech capacity; the writers were fully aware of Moore's law.)
This apparently doesn't extend to Data, who once measured his capacity in bits.
Was there a reason for this or was this an oversight?
The machinery and software comprising Data and the Enterprise computer are different, and so the terminology is separate.
Unfortunately there's not a lot of information about the various hardware, software, or languages upon which either LCARS or Data functions - we know Data's brain is positronic, and that the ship implements isolinear chips, but what that means for their construction and purpose is not well-explored.
They also play a little fast and loose with their 'computery-sounding words'. For example - the show throws about the word subroutine an awful lot, which is probably technically fine, but even by 21st century standards, the preferred nomenclature for languages at least as 'high level' as C tends to be function, or if it is bound to an object, method.*
It is unlikely that Data is writing his new subroutines using Java or Kivy, despite its compatibility with Android - so he may indeed self-update in native machine language. LCARS users, however, I would expect to have a spectrum of languages available to them which allow for various abstraction layers, similar to today. There may well be a C equivalent, a Python equivalent, a PHP equivalent (Q forbid…), and so on. As for the languages and hardware implemented on either 'platform' - again, it's just lost in the details or lack thereof.
For what it's worth, bytes have been applied to human brains, so it offers a basis of comparison. At present it is estimated that the human brain represents 2.5 petabytes of binary data storage. Data, having a stated storage capacity of 'eight hundred quadrillion bits', puts him in the ballpark of 100 petabytes. If it is meaningful to measure the human brain in this way, then it conveys something meaningful about Data's brain in relation to our own - it represents roughly 40 times as much raw storage capacity. So this is one potential reason why he may have expressed his storage in this way; in particular, given the circumstances of the trial, finding a common unit of measurement was a good move on his part.
Our current technology doesn't make these distinctions, but then again they are arguably far more similar to each other than the Enterprise computer and Cmdr. Data. All our technology comes from the same planet; our phones need to talk to our computers, and increasingly our computers can or should be used to automate at-home tasks, so everything we currently know about computer science is driving convergence of both hardware and language, to facilitate compatibility. In Star Trek, this level of inter-communication and compatibility is arguably hugely insecure, as illustrated by Kirk's use of the Reliant's prefix code in Wrath of Khan (what is that, anyway, like a port number or something? eesh). The computers may need to be measured in quads out of necessity, as they should not be so readily read and understood by every passing ship that means to glean information from their computer. So the hardware and software are unique to their own design, and in this way a Ferengi can't just camp under the Enterprise and steal their wifi. That's entirely conjecture, of course.
*I hope not to derail by using such terms, or to have it endlessly pointed out wherein the terms subroutine, function, method and so on are or aren't interchangeable. I do enough scripting to know that you get funny stares for calling things subroutines, and if programming LCARS is as trivial and wide-spread as TNG makes it appear, I personally find it a little strange that the popular language would regress back to subroutine after adopting terms and practices such as writing functions.
As a computer programmer, I can give you one possible explanation - memory naming nomenclature.
A starship is inevitably going to have more memory storage than a biometric android, no matter how complex Data may be (how he would manage to store the Enterprise's subroutines on his mind in "A Fistful of Datas" notwithstanding). So the simple answer is - the difference in scale of memory storage is simply great enough to warrant different units of measurement.
I had always imagined a 'quad' is a four-logic operator, like a 'bit' is a two-logic operator. A quad, therefore, is like a bit, but it can exist in one four possible states, instead of merely two. I'd imagine you'd need a quantum computer in order to have a switch be able to turn to two options other than merely off and on, such as off-and-on superimposed, although that would be a three-logic operator, and I can't think of what a fourth state could be.
But Data measuring his storage in 'bits' throws the whole thing for a loop.
Realistically, it's probably just technobabble the writers used to say, "this computer's really advanced!", so they'd have an excuse to make it able to perform any feat they could imagine, and the bit thing with Data was an oversight.
Star Trek's full of stuff like that. You should Google, "A wizard did it."
Don't forget that Data is a person, or at least sees himself as one. Where references to the capacity of the Enterprise are read from a list of technical specs, Data's answer reflects his personality and the situation.
It may be that Data chose to use the archaic terms "bits" because it wouldn't be as familiar to the judge as the modern term "quad". Maybe he thought it would make him sound less like a computer.
