An acute disease can sometimes recover naturally, but always with a cure.
A chronic disease can't recover naturally and there is no cure.
Is this the difference ?
More or less: Yes.
Acute and chronic are ways to classify diseases according to duration.
Acute*:
Acute Illness (1) Any illness that develops quickly, is intense or severe and lasts a relatively short period of time. (2) Any condition—e.g., infection, trauma, fracture—with a short (often less than 1 month) clinical course. Acute illnesses usually respond to therapy; a return to a state of complete—pre-morbid—health is the norm.
from: Segen's Medical Dictionary. © 2012 Farlex via The Free Dictionary
Chronic diseases are different. They usually develop slowly, last a long time, and are often progressive and incurable. For many chronic diseases, there is no cure. The long-term effects of a chronic illness may be difficult to predict.
from: Island health
Note that most, but not all chronic illnesses are incurable. Also, although the illness is present all the time, the symptoms don't have to be. The time periods (which can be quite long) when the person doesn't have symptoms and feels well is called remission. When the symptoms appear this is called exacerbation or relapse. Many chronic illnesses that can't be cured can be well-managed and the consequences can be mitigated.
There are other categories between chronic and acute:
subacute - which can refer to duration (between acute and chronic, but closer to acute), to severity (often duration and severity combined: moderate) or it is sometimes used to denote a condition in a person who appears to be clinically well (although there are better terms for this, IMO).
subchronic - usually longer than acute, but limited in time (duration of illness is for about a year).
Subacute and subchronic (along with acute and chronic) are also used in toxicology to refer to classification of toxicity according to duration of exposure.
There is no universal duration time for all illnesses, which would serve as the limit based on which the illness would be acute or chronic. The duration based on which conditions are classified often vary across different fields of medicine (i.e. different organs/organ systems). Consider these two examples:
Acute sinusitis - Duration of illness less than 3 weeks.
Sub acute sinusitis - Duration of illness ranging between 3-6 weeks.
Recurrent sinusitis - Recurrent sinusitis with disease free periods in-between attacks.
Chronic sinusitis - Duration of illness is more than 6 weeks
from: Role of Imaging in Rhinology edited by Geetha Ramamoorthy
Cases of low back pain can be classified according to duration as follows:
Acute: less than 6 weeks
Subacute: between 6 and 12 weeks
Chronic: more than 12 weeks
from: Integrative Medicine by David Rakel