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.
The best channel to use is the lowest channel that has a high transmit power. Lower frequencies penetrate walls more easily, and there is a significant difference between channel 36 (5180Mhz) and channel 165 (5825Mhz).
However up until 2014 there were limits on transmit power for the lower frequencies, so depending on your router you may need to use a higher channel to get larger transmit power.
So in reality you should just try channels within the various 5Ghz bands and see which gives a better signal. Then choose the lowest channel within that band.
Your “better” router probably supports that fuller list of channels as well, but it doesn’t show you those channels for manual channel selection. As @YLearn’s Answer mentioned, those other channels (52-144) require DFS, which means that your router might not be able to use some of those channels even if you had picked them.
It’s confusing to users if you let them manually pick a channel, and then you end up having to use a different channel anyway. It makes users feel like you lied to them when you let them manually pick a channel even if you knew you might not be able to honor their choice. And you sort of did lie to them. So better router vendors only let you manually pick the channels they know for sure they’ll be able to use. To access any other channel, you just have to rely on automatic channel selection.
To answer the other half of your question, channels 149-161 will provide better range than 36-48. The higher band broadcasts at a stronger signal than the lower band (I believe this is due to the standard).
If you are in a house where your wifi needs to penetrate multiple walls and/or floors – choose a low channel. If your in a loft apartment or want your wifi to extend into the back yard with a near clean line of sight, choose a higher channel.