The channels that your "better" router has omitted are what is known as the dynamic frequency selection or DFS channels. These channels have been allows by the FCC for use, but they must employ a means of detecting and shifting off the channel if they detect another licensed use of the spectrum (namely some forms of weather radar and very limited military use).
Many enterprise deployments will avoid these channels if possible for the same reason.
It sounds the like the vendor of your "better" router has decided to avoid those complexities and simply ignore that they exist. This still leaves you eight non-overlapping channels which is five more than you have in 2.4 GHz (which has only three non-overlapping channels) and far more than you need in any residential/consumer deployment.
As to which to use, download a copy of inSSIDer and pick the one where you see no or the lowest signal strength in use.