If each cell could store a discrete bit of info, then our craniums have room for about a trillion bits. Hardly impressive.
An arbitrary unit as a 'bit' per cell seems very low, especially since it has been recorded that the brain uses many different chemicals all over the place to store information -- the capacity should be significantly more than just a bit per cell. Researchers also found that certain stimuli will only activate a few dozen brain cells to 'deal with it'. A lot of very complex things are going on on a ridiculously small scale.
In addition to that, in many ways the brain does act as a large database -- rather than storing the same 'objects' over and over and over again, it appears that in many cases the information stored in your long-term memory 'references' other memories. If one gets damaged somehow, it can affect many other memories with the same false information. You also make a (subconcious) selection of what information to store -- your brain doesn't store full motion video at 100fps day in day out. Much information deemed insignificant will not be stored in your permanent memory at all, and many other things will be very fuzzy -- you can recall the basics, but not the details. Nice garbage collection system, too.
A human brains start out with about a thousand trillion synapses at birth (connection points between the various signal cells), and estimates for adults range between a hundred trillion to 500 trillion. How many data lines & connection points do the current generation of computer chips have?