This discussion takes place about once a day on various forums.
I always advise booking directly with the airlines, never with OTAs. This is because I prefer the airlines to be able to price discriminate between market segments, with a steady stream if higher revenue direct booking pax, leaving lower prices for people like me who book "risky" flights via OTAs.
On a less serious note....
Buying tickets on 3rd party OTAs (as long as they are reputable) has several distinct advantages:
1. The price is often lower, sometimes significantly so (for the same fare class, i.e. "apples to apples")
2. Multi-carrier (cheaper or more convenient) itineraries are offered which would be difficult or impossible to find on an airline website
3. For non-US flights you have the benefit of a 24+ hour (sometimes up to 48, depending on when you buy the ticket) cancellation window, this can be quite significant if you have other flights/hotels/etc to book and need the option of cancelling if something falls through
4. There is a (small) cashback in the form of points/orbucks and the like
The disadvantages are that the airline may be less flexible when it comes to changes - however the likelihood is that booking via an OTA you will be booking inflexible nonrefundable tickets in any case, so even if booked direct via the airline flexibility is close to zero.
I fly roughly 100 legs a year, with roughly 2/3 of my flights booked via OTAs. While it is true that airlines (and car rental agencies, and hotels) are less flexible when it comes to itineraries not booked directly with them, I have not found the difference to be huge. In some cases I felt I got better service from the OTA.
As to "when things go wrong" I have been through cancellations, weather delays, reroutings and so on - was treated the same as any other pax (got good service form good airlines, poor service from poor airlines....).
Having said that, with US domestic travel, though, OTAs seem to usually offer the same price as booking direct. And airlines are obliged to offer 24 hour refunds - which negates the two main benefits of OTAs.