Why would a company want me to book a flight via a travel agent if it is more expensive?

There are several reasons why such internal policies exist:

• Your institution may get a discount on certain services of the travel agent if it buys enough from the agent
• By having a contract with a reasonably-priced travel agent, they make sure that no employee books the flight with a very pricey agent or with a travel agent that happens to be a relative of the traveller
• The travel agency offers easy methods to verify expense information and thus may lower the workload of your local administrative personnel. This especially holds if the payment is routed directly from the institution to the travel agent, which also makes sure that the flyer does not get credit card bonus points for the booking.
• Your institution may have an agreement with the travel agent that they manually check if members of your institution are not unnecessarily booking expensive booking classes (in order to get more airline bonus miles)
• Some insurances that your institution bought may only work if the ticket is bought directly from the institution and not by the flyer, and they elected the agent to handle such bookings.

Which of these reasons is the one why this policy exists can only be answered by the responsible employees of your institution.

All the answers so far seem to miss an important point: Your university might be legally required to do that. Specifically, above a certain amount of money, in many countries, public sectors organisations have to go through a specific procedure (call for tenders) to procure goods and services.

Something not mentioned so far is the fact that the ticket booked through the travel agent may be refundable, where the ticket from the online vendor is most likely not refundable. A number of years ago when I traveled on behalf of a government agency this was the case. We often booked flights that were one or two hundred dollars more than the cheapest price you could find, but the travel agent guaranteed the price for flights on short notice, guaranteed the ability to change the flight itinerary, and guaranteed a refund if the agency had to cancel. To the government all this was worth the extra price.

On top of the other answer by DCTLIB...

You haven't specified where you are and where is 'your' company, but there might be tax or financial book-keeping reasons.

E.g. if the company is in EU they might require a proper VAT invoice to properly book it (and maybe get a VAT refund). And some vendors, especially in US, are unlikely to provide that. Using a travel agent might give the university confidence in receiving proper paperwork. $50 is likely a saving not worth enough to chase for that reason.

Another reason - they think the time you would be spending looking for cheap flights is worth more than the price difference.

Large travel agents have 24/7 assistance numbers. If you end up stranded at 2am on Sunday, they can get you rebooked, find a hotel room, etc. Most organizations are simply unable or unwilling to provide that support. OTA's (online travel agents) such as Expedia and Travelocity are notoriously bad at pointing a finger at the airline or hotel, while the provider correctly points back at the OTA who "owns" your booking. Lastly, when money is due back to the organization, it can be difficult and time-consuming to recover through the airline or an OTA. Large TA's that handle corporate accounts have the clout to handle credits and debits.

At many universities this is recommended but it is rarely a requirement, for the reasons listed in the other answers. However, there are downsides to booking via a travel agency, e.g. if you need to rebook the flight back while you are at the conference. You then need to contact your travel agency to do that for you, and they may ask that you make a payment for the rebooking first, putting the rebooking on hold until the payment is received. So, it's actually quite risky to book a flight via a travel agency, which is why I never do this.

IF in UK, something called ATOL protects all purchased tickets.

IF in a backward place like the usa;

• One suspects that, in the case of 'a problem', they know which building to burn down and who gets a visit from 'BigVinnie' and the 'boys'.
• Additionally, someone's brother-in-law is probably involved somewhere...