What is the difference between an acute and chronic sports injury?

Hi there, Great question.
The short answer is:
An acute injury is an injury that has recently happened.
A chronic injury is an injury that has gone beyond the expected time for healing and/ or a recurring injury. I hope that helps. Monte

I’d just like to expand on Monte’s answer for those who’d like to know more about additional features of acute and chronic sports injuries.
 
In addition to being recent, an acute injury may be due to the body having received an external force – such as a fall, collision with a competitor/opponent/object – or even be due to internal forces of the athlete’s own body.  If the external force is great enough, normal healthy tissues can be damaged – eg. bones, muscles, tendons, ligaments.  Alternatively, an external force of lesser degree can cause an acute injury to an athlete’s tissues that were already weakened for a range of reasons.  In contrast, all that is required to set off an acute injury due to internal forces is certain postures and movements of the athlete alone.  This is an interesting phenomenon, which can catch the person by total surprise, or can be preceded by milder ‘niggling’ discomfort for a time; which in hindsight to the individual would be considered to be the ‘warning signs’.  So, if any particular body tissues have become weakened or compromised such that they are not able to do their job properly, not much is needed to tip them over the edge into acute injury territory. 
 
The duration of an injury varies according to the time necessary for different tissue types to heal – eg. typically contusions/bruising heals quicker than sprains/strains, which in turn heal quicker than a shoulder dislocation.  There are two stages to the healing process in a physical injury – acute and sub-acute.  In the acute stage, inflammation is the process the body uses to get healing underway and repair the injured tissue/s.  Following this, the tissues gradually get stronger during the sub-acute stage until they have fully healed. 
 
As Monte mentioned, an injury is regarded as chronic when the athlete does not recover from an acute injury in the typical timeframe for the particular injured tissues.  This could be because the injury is complicated, such that only partial healing is possible, or that full healing takes longer than usual.  Sometimes an acute injury actually heals, only to recur on an occasional or frequent basis – which can be quite frustrating for the athlete.  Unfortunately, in other cases there is complete healing of a physical injury, but the person continues to suffer ongoing pain.  For more information, see the Healthshare answers to: What is chronic pain?  and: What is the best way to manage chronic pain?
 
Recurring injuries frustrate a person’s enjoyment of their sport, decrease performance and regrettably, can even result in them having to cease sport altogether.  When a relatively small external force and/or internal forces of the body are all that is needed for an acute injury to occur, a recurring cycle of symptoms is the common result.  Damaged structures, such as a meniscus can lead to a swollen knee, or a lower back disc can lead to a grumpy back whenever the athlete loads up or moves these structures a certain way during their sporting postures and movements.  In other situations, less than ideal quality of postures and movements during training, competition and various life activities means that the body operates with imbalances.  The development of these posture and movement ‘faults’ may occur from training technique/equipment errors over time, particular work/life activities, due to having not developed ideal postures and movement patterns as a baby, other health problems.  However the faults are caused, inefficient postures and movements make an athlete more susceptible to injury compared with those athletes having efficient postures and movements.  Furthermore, injuries are also more likely to occur when an athlete is tired and/or lacking sufficient aerobic fitness, such as after a recent acute injury episode or other illness.  This is because it is harder for them to maintain efficient postures and movements at higher levels of activity intensity and sport skill.
 
While not all acute injuries require assistance from a sports care practitioner, skilled clinical care and advice from one or sometimes more practitioners is necessary for complicated, complex, recurring chronic injuries in order to obtain the best outcome for the injury.

The short is answer is like Monte described above. 

An “acute injury” means it has recently happened.  Chronic pain or injury is typically when someone has experienced a probelm for 6 months or more. 

Then there's a subacute injury, it's in between acute and chronic - from around 3-4 weeks to 6 months.

A McKenzie Physiotherapist is able to assess your pain and determine the best exercises your need to fully recover.  In the acute phase, you just need to follow the RICE principles:
Rest
Ice
Elevation
Compression

Most people will know an acute injury as it happens. Generally speaking you will experience a sharp and sudden pain at the time of injury i.e. ankle twisting during football. More often than not the injury will result in pain, swelling and/or loss of function i.e. unable to weight bear or weakness. Once an acute injury occurs, you should stop what you are doing to avoid further injury. Apply RICE principles. Have it assessed properly if the severity of pain and swelling becomes too unbearable. You may need an extended peiod of rest from your sport whilst the injury heals.

Chronic injuries  are ones that you have had for at least 3 months ongoing. Typically it may have started as an acute injury and the pain became more and more bearable with time, or it was never completely resolved before getting back on the field. This happens quite often with high level athletes as the pressures from external sources i.e. club, coach, team, sponsors etc demand you return quickly rather than spending extra time on the sidelines. Quite a controversial topic amoungst healthcare providers. Chronic injuries, may have even started as a little 'niggle' that only appear occasionaly. It may progress with age and time as always being there. Particulalry if it is not managed or addressed correctly by a healthcare provider. Chronic injuries may even be a result of unsound biomechanics which result in premature wearing of the tissues. Over time as you continue playing and more wear and tear accumulates, these injuries become more noticeable. You can also have an acute flareup of the chronic injury. These are more common in the recreational populations.

Unfortunaely the longer you leave these injuries, it can mean that there is more difficulty treating them, as the quality of the tissue becomes less healthier over time. You may feel that you can continue with your sport and your performance may be affected only slightly, but it is recommended that you stop/modify your activity until it has been properly assessed by a healthcare professional.

Either acute or chronic injury requires a proper assessment and quite often time away to recover from the injury.