Introduction to an Ageing Population (Greying Population)
Before going straight into the specific advantages and disadvantages of an ageing population, also commonly referred to as a 'greying population', I believe it is first best to gain a common understanding and concept of what an ageing population actually is and how it occurs.
The term, 'ageing population', can be said to describe the demographic transition from the young to the elderly as a proportion of the total population (i.e the average age of the population increases). If you refer to population structure diagrams, you can see this visually from the formation of a 'pyramid' type shape to that of a 'kite' shape.
An ageing population is primarily caused by two demographic drivers, one of a decline in fertility rates and the other as a result of an increase in life expectancy. This process has never been encountered before in the entire history of mankind, therefore when discussing the following advantages and disadvantages of an ageing population it is important to bear in mind, they are mainly guesses and estimation, no one really knows the true outcome (pros and cons) of an ageing population yet, as it has not actually 'fully' happened.
To gain a deeper and more extensive grasp of the concept of an 'ageing population' and its advantages and disadvantages, I personally suggest you read books such as 'Shock of Gray: The Aging of the World's Population and How it Pits Young Against Old, Child Against Parent, Worker Against Boss, Company Against Rival, and Nation Against Nation' and 'Rethinking Aging: Growing Old and Living Well in an Overtreated Society'.
Advantages for an Ageing Population (Greying Population)
Growing Pool of Volunteers - the argument undertaken by Judith Healy in her paper 'The Benefits of an Ageing Population' (2004) argues several benefits that accompany an ageing population and one is that of the advantage gained through the increase in the number of volunteers. Healy argues several merits from volunteering one of which she includes as "Volunteers contribute a substantial amount of unpaid work in terms of person hours to a variety of community and public sector agencies" (p.24). With an estimated 30% of 65-74 year olds volunteering (in Australia) and with further government intervention for greater percentage to be active in volunteering after retirement - society can potentially gain both economically and socially through a new 'greater' influx of volunteering and essentially 'free' economic and social input.
Economic Savings - in most cases, although the population is getting older, this generally means however the country has less children and therefore they (the country) should be making savings on the costs 'that would have been' if the country did have all those children. Hence, therefore posing as a potential positive to the opposite negative of an increase in costs from welfare with an greying population. It has been quoted that Japan in particular has saved millions on the potential children that could have been in terms of educational costs and healthcare costs. 
Increased Social Harmony - there is a general consensus that although there is the possibility of 'intergenerational warfare' (although this is mainly disputed in economic literature), there is also the possible positive factor of increased harmony across society. For example, as Healy argues, it could make a society more 'law-abiding'  as there has been proven to be a lower correlation of criminal activities and aging (positive of an increase in savings from criminal related costs). Moreover, as a result of more people retiring, this allows for the increase in job vacancies and as experienced in Japan, this constitutes towards a rise in employment levels - resulting in increased prosperity etc. therefore likely discouraging unsociable activities.
Overview of the Pros and Advantages of an Ageing Population
Disadvantages for an Ageing Population (Greying Population)
Increase in Pension Costs - one of the most common and dramatic concerns and drawbacks of an ageing population is the likely increase in public pension costs to a government and therefore the public. The increased pension costs problem (regularly dubbed as 'the pension crisis') is as a result of more and more people retiring, and hence there being a greater and greater drain on resources to fund the ever-growing pension costs. The pension costs occur from two points - firstly people are living longer meaning they can withdraw pension funds for longer and secondly there is simply more pensioners.
Increase in Healthcare Costs - the additional main concern and potential negative impact of an ageing population is that of the 'health cost scare'. As a result of 'older people' generally being more susceptible to being more prone to illness and where medical care is provided by the state, then this will therefore, as a result of an increase in the amount of older people, will cause the healthcare and welfare costs to increase. There is however an additional question asked - Whether people will stay ill for longer or will the illness just be prevented for a later stage within their life? If it is the former then we are likely to face even further additional welfare costs, as like the pension point above, two points of cost can be drawn that of: medical costs must be maintained for longer and simply there is a greater number of the elderly.
Decrease in the Support Ratio the support ratio is defined as "the ratio of dependents--people younger than 15 or older than 64--to the working-age population--those ages 15-64". Therefore, a decrease in the support ratio translates loosely into there being less workers and therefore less tax contributors in the future for the 'increased' welfare costs of the future. Hence, a double-edged effect occurs. Not only are there less tax contributors, but there is likely to be an increase in the amount of tax that is needed.
The Consequences of an Ageing Population
Ted Fishman Author of the 'Shock of Gray'
Summary of the Advantages and Disadvantages for an Ageing Population (Greying Population)
In truth it is difficult to exactly pinpoint whether such advantages and disadvantages for an ageing population will occur along with their extent and magnitude (and its merits and demerits may perhaps be different to every country). For example, no one can really determine the extent as to which healthcare costs will increase (if at all - through the compression of morbidity vs expansion of morbidity debate) and then to factor in the account of potential growth, coinciding with the increases, makes the task even more difficult. No one can really account for future economic growth (which may allow us to simply be able to bear the cost) and future technological advances - who knows perhaps health care costs will come tumbling down due to new mechanical nurses. So even if we can state 'x and y' are the problems that are going to happen (which we can't exactly), we do not really know what economic situation in the future we will be in to deal with the issues.
If you have any additional advantages and disadvantages for an ageing population ('greying population) or any general insights on an ageing population, as well as any general remarks, then I encourage you to please leave a comment below.