It is common for people to think fighting is sport and vice versa. It isn't, although they have some things in common. Sport can build some good attributes which can be beneficial to surviving a fight.
Sport practice and training builds fitness, strength, fluid movement, speed, balance, and overall awareness of what your body can do. One of the best things is the hand-eye coordination you get from sport. Almost all athletics provide these benefits to some degree or other.

Martial sports give you, again to one degree or other based on which martial sport we're talking about, experience with watching and reading an opponent (a live opponent who is in a fight with you).

Here's where the differences come into play.
1) A sport fight is announced so you know exactly when it starts. It is never an ambush. Both fighters know exactly what is coming and when. In self-defense, ambushes are very common. They are also often initiated with what is called an interview, which is a predator eyeing up his prey and approaching to see if the prey will be an easy mark or not so he can decide whether to attack.
2) In a sport fight, you are almost always given advance notice of who you are going to fight, which gives you the chance to study and analyze your opponent. You find out his strengths and weaknesses in advance so you can form a plan for how to deal with him. You work this into your training for weeks or months in advance of your fight.
3) Almost all sport fights have rules against injurious behavior, which is understandable. A street fight is often times out of anger or rage and causing injury to the other person is not a concern, in fact it is a priority. I'm not just talking groin kicks and eye poking but pulling a knife or other weapon, having a others join in (probably to your detriment) are commonplace in street fights. Sport fighters find out the hard way that their training lacks these things because they never need these skills in a sport fight. A sport fight is about a fair competition. A street fight is about overwhelming your opponent no matter how underhanded you have to be.
4) In a sport fight you will almost always face someone who is in about your weight class, about your skill level, and trained in similar skills you are. The terms of the fight require that both parties agree to the rules, see #3 above. None of these are assured or even likely in a street fight. It is quite likely you will be outmatched in weight, strength, speed, or skill. Most people who start street fights are confident they will win, and have some kind of ace(s) in the hole making it likely they will be the one to come out on top. 
5) A sport fight is always well lit in a clear open area with clean floor which has a consistent surface. A sport fight is never in a stairwell, parking ramp, bathroom, alley, in the rain, on wet or or beer soaked floor, etc. These are all VERY important factors. Also, sport fighters don't fight once they've drunk a half pitcher of beer or a handful of cocktails either. That makes a HUGE difference.

A core concept of security is that if you seek to protect yourself from a threat, you start with a threat profile. That is, identify the type of threat you are expecting and form a plan to deal with it.

Bodyguards and security personnel do this all the time. Someone who was threatened by a gang is an entirely different threat profile than a jeweler who has to transport high value merchandise, for example.

Self-defense is no different. What type of physical threats are you going to be dealing with? What type of environment are you going to be in? What degree of preparation are you ready to commit to?

There are many more questions to address when it comes to deciding how you are going to proceed with forming your security plan and how you will train to fulfill it.

Unfortunately, martial artists tend to fall into the trap of wanting to learn techniques thinking that is all there is to know about surviving a violent encounter. There is MUCH more to it than learning how to get out of a headlock or block a punch. These are useful for self-defense, for sure, but the art of self-protection is a whole nother realm.

Another aspect of this trap is thinking that sport fighting training prepares you for real world violence. Admittedly, being strong and in good physical condition is a good thing - a very good thing. Without good self-protection skills they likely won't be nearly as useful as you would think. You should definitely not use this as an excuse not to be in good physical condition or get lazy in your conditioning.

A good martial artist is a balance between skill and physicality.