There is an important distinction between being "sober" and being in "recovery". The differences are clear to those who have experienced both phases in their healing process from alcoholism.
When an alcoholic is "sober" from alcohol without attending a mutual-help program, therapy, medication management and/or treatment then they are in a sense "white-knuckling" their sobriety (also referred to as being a "dry drunk").
These individuals may be staying away from alcohol, but they are not treating the underlying issues that had either led to their drinking in the first place or developed as their alcoholism progressed.
Many "sober" alcoholics who are not in "recovery" will experience a transfer of addictions that could involve a new addiction to food, sex, shopping, romantic relationships, etc. because they have not found a healthy way to fill the void that alcohol had satisfied. They may have stopped drinking, but their life may be exactly the same, leading them to be jealous of others who are drinking or to struggle with emotional or mental health issues.
An alcoholic who is in "recovery" is essentially in remission from alcoholism. Their alcoholism is not cured but is at bay in a way that allows them to be free of the cravings and mental obsession.
They have also treated their underlying issues (mental health, spiritual, physical) that led to or resulted from their drinking. These alcoholics have found a way to fill the void once satisfied by alcohol through spiritual, emotional and/or behavioral solutions that they have learned through treatment, therapy, medication management and/or mutual-help groups (A.A., SMART Recovery). They have made significant changes that have allowed them to find peace in removing alcohol from their life and to have emotional stability.
Some individuals with drinking problems are able to stop drinking effortlessly and permanently without craving or obsessing about alcohol. They may not have been alcoholic in the first place, but instead are heavy or problem drinkers (see my past post on differences between social drinkers, problem drinkers, and alcoholics for more information). In contrast, alcoholics may abstain for periods of time without help but in most cases will inevitably return to their previous drinking patterns.
Alcoholism resources and more information on the topic of high-functioning alcoholics are available here.