Q. On a recent rainy day I was searching for flights to Christchurch, New Zealand. When I checked roundtrips departing from O'Hare, the cheapest fares ranged from $1689 to over $2300. When I searched roundtrips from Los Angeles to Christchurch, I got prices from the mid $900s to $1042. The difference in departing from O'Hare versus from LAX ranged from a minimum of over $700 with Travelocity to almost $1300 with Orbitz. I could fly from O'Hare to LAX and from LAX to Christchurch for far less than flying all the way on one ticket from O'Hare. This strikes me as odd. Could you explain?

I would prefer to travel on a single ticket since I think I would be better protected in case there are complications getting out to LAX. However that approach seems expensive, particularly since I have enough miles on American, Delta or United to have a reasonable chance of getting out there on one of them. Another complication is I'm taking my bike along (that would seem to prohibit using Delta with their exorbitant prices to take a bike). What are the drawbacks of having separate tickets ie; a roundtrip from Chicago to Los Angeles and a roundtrip from Los Angeles to Christchurch?

A. The price difference is probably due to the difference in the demand and/or supply of the local airfare market.  Simply put, there are probably more flights available to choose from to get to New Zealand from Los Angeles, which creates greater price competition in this market and results in lower prices.
You are right about booking a trip as two separate itineraries.  Any protection due to delays or cancellations of flights will not apply, so that's a major consideration. If you miss a connecting flight, the airline won't be required to rebook you on another flight, a potentially expensive catastrophe. Also, you may not be able to check your luggage all the way through (including your bike!) and this would create an additional hassle during your travels.  You may have to retrieve your luggage and check it in again, so do some research to find out what to expect.  And be sure to check for both your outbound and return, because sometimes inter-lining agreements between carriers can vary, depending on the route and direction you're going in.
Of course, saving money is always a good thing...but sometimes paying more for a little peace of mind can be worth it.