Yep, it's pretty much all about price, although the highest fare codes usually are a bit more flexible with regard to changes and refunds. Otherwise, the only difference between different fare codes (aka "buckets") is usually the miles they earn, since many airlines give only 50% miles for the cheapest fares.

LY, like other airlines, use the different buckets to manage revenue on each flight. They will release the cheaper buckets on a given flight if they feel there's not enough demand for more expensive fares, and as the cheaper buckets sell out, pax have to pay more to get on that same flight because only the more expensive buckets are left.

The travel agent should be able to see the fare code at the time of booking. A complete fare code is a string of letters/numbers where the first letter is one of the buckets. Examples include "NSPRING" (an "N" fare) or "HJTLV" (an "H" fare). You could probably get a more expensive fare by asking for it (if a cheaper bucket is available, a more expensive one usually is, too), but the agent might be confused by the request, since the vast majority of pax want the cheapest fare available for a given flight--except for those who need flexible/refundable fares. Before airlines started differentiating miles earned based on buckets, I don't think most people paid any attention to buckets (except for the refundable/nonrefundable distinction).

Does that answer your questions?