For context, I ride in the North Shore of Vancouver, BC, Canada, which is an area famous for its steep and technically challenging trails. I'm comfortable riding black diamond-rated trails. Here's an example of one (not my video):
Personally, I ride SPD. I've tried Crankbrothers for six months, but I didn't like the feel, and I wanted more tension. Clipping out with CB is akin to walking up a ramp, where the resistance smoothly and gradually increases until you suddenly reach the release threshold. In comparison, SPD is like jumping up onto a ledge; there is no resistance whatsoever until you hit a definitive "wall" of resistance right before you clip out. I like to be in control of my clip in/out and didn't like the ambiguous feeling of CB. Unlike Shimano, CB also lacks a tension adjustment.
I almost never proactively clip out. I rarely ride in loose enough conditions where the foot out technique is greatly beneficial. I'm always clipped in unless I'm crashing or losing my balance during a technical climb (known as a "dab").
At the recreational level, riding flats or clips on mountain bikes is mainly a matter of personal preference. If you feel better riding clipped in, go with that. If you prefer riding flats instead, then you should do that.
Some of the benefits:
- Foot stability. Especially with the stiffer soles typical of clipless shoes, your foot has to do less work to keep itself rigid and held in place, especially during repeated heavy impacts.
- Sprint power. MTB involves a lot of sprinting, and the increased number of muscles you can engage by pulling up when riding clipped in helps significantly.
- Pedaling smoothness. This is not critical for downhill-focused riders, but if you're interested in riding technical climbs, you can get a smoother pedal stroke with clips which will help you to not lose traction.
- Foot retention. This is the obvious one; riding clipped in means your feet are less likely to inadvertently fly off the pedals.
- Pulling up. Although this should only be used once proper weight transfer skills are learned, riding clipped in means you can "cheat" and pull up occasionally instead of having to do a full body movement every time you want to hop/ride over a small obstacle. This is more efficient and saves you energy.
- You get better control over the rear end of the bike as detailed in Criggie's answer.
- Foot retention. This is one of the biggest sticking points for clipless beginners, who are often frightened by the prospect of being mechanically latched onto their bike during a crash. By training your reflexes, you can clip out during crashes no problem, but it is ultimately going to be harder to some extent to clip out than it is to simply step away like you can with flats. This can be a major detriment towards confidence. Sometimes, riders will not try to ride a challenging feature because "I'm clipped in".
- Skills development (mostly for beginners). Being clipped in means you can procrastinate on learning weight transfer skills, which are mandatory. Instead of being forced to shift your body weight to perform maneuvers such as bunny hops and riding jumps, you can just pull up on the clips instead. This façade will be inadequate someday and you will immediately realize that you are unable to progress as a rider because you lack those basic skills.
- Makes riding the bicycle more complicated. Now you have to put on special shoes every time you want to ride, and you will have the additional cognitive load of controlling clip/unclip decisions. Additionally, your non-cyclist friends will struggle to try riding the bike and you'll have to explain the clipless pedal system to every single one of your (curious) friends, family, spouse, children...
- Fixed foot placement. Current clipless shoes don’t allow you to adjust the cleat position on the fly. This means that you might not be in the best foot position for every situation; for example, you might want the pedal spindle farther forwards during a climb or sprint for more power, while you will want the spindle farther back while you are descending. Flats allow for this flexibility. Another factor is how many pairs of shoes you own: I can only afford one nice pair, so I have my cleats slammed all the way back for MTB riding, which is a position that isn’t always best for road riding. Flat pedals would allow me to place my foot wherever I want.
Of course, your local trails are a factor. Someone whose trails are long, mellow singletrack routes may benefit from riding clipped in, while someone whose bread and butter is absurdly steep technical features or jump trails might have a bias towards flats.
At the professional level, the benefits of clipless in terms of power transfer outweigh the confidence-related side effects because the riders are all extremely experienced. They're not going to be scared of riding something just because they're clipped in. There are notable exceptions to this generalization though because of people who use the confidence boost to ride faster than having the power benefits could.
And again, the type of riding needs to be considered. Hardly anyone rides flats at XC races, while the opposite is true for dirt jumping or other stylish jump disciplines. If you plan on racing or doing any other kind of competitive riding, I would recommend trying both clips and flats for a decent time each (several months minimum) and doing timed runs as a comparison.
Of course, there is nothing wrong with learning how to be proficient and confident in riding both pedal systems either.