Pretty general question -- maybe a couple pros and cons.
Pros of landfill. Most new landfills are built as lined facility, with leachate collection and treatment on-site. Because the landfill is designed and built to store waste, this is thought to be a safe containment of our garbage.
Methane gas is collected at a number of municipal waste landfills and some landfills have diesel/gas blended fuel fired electrical generation stations built on them. You can research some of the work recently conducted by Ingenco (Ingenco.com). Therefore, a "free" source of fuel to generate electricity.
Upon closing of a landfill, some beneficial usage of these land resources have been realized. Certain closed landfills have become golf courses, ski mountains, etc.
Cons of Landfill. There are landfills that are currently utilized that are unlined. There are countless landfills that have been closed that were not built with the same level of scrutiny that our newer facilities have (unlined, minimal or no leachate collection and treatment).
There are countless hazardous wastes that are blended into the municipal waste stream. Ie. lead from batteries, mercury from light bulbs, heavy metals from tv and electronics, etc. When these hazardous waste steams find there way into municipal waste landfills, the contaminants can find their way into the aquifer and into our drinking water.
Landfills are typically very unatractive. Landfills are often constructed in low lying areas and built in cells that may encompass 50 to 150 acres (or more). They attract varmin, birds, bears and other not so attractive wildlife. They can have some odor that can often be detected outside the bounds of the property borders.
The traffic around landfills can be significant -- especially during morning and early afternoon hours. Because landfills are built in areas that are less populated, roads are often not equipped for this increase in heavy traffic. Significant infrastructure improvements are often required.
Upon landfill closure, land use is extremely limited. Plus, there is always a risk of methane production -- which can be flammable (therefore, open lights on landfills are often not permitted).
Incinerators are typically not called incinerators. They are resource recovery facilities (RRF) or waste to energy facilties.
Pros of RRF. When waste stream is brought to RRF, most RRFs sort the waste stream. Therefore, more recycling of the waste stream takes place.
RRFs reduce the amount of material that must be landfilled by more than 90%. Hazardous streams are more easily removed from the waste stream because of the additional sorting.
RRFs require a significantly smaller footprint than a comparably sized landfill.
The waste that is "incinerated" is used to generate heat -- this heat is used to heat neighboring buildings (or neighboring tenants) and high quality steam (for electrical generation). Therefore, the trash is used as a free (actuall a fee paid) source of fuel.
Cons of RRF. When incinerating municipal waste, certain pollutants will be emitted to the air. The EPA is concerned with the following pollutants -- Particulate (and PM10), Carbon Dioxide, Carbon Monoxide, Metals (including Hg), Hydrogen Chloride, Dioxins/Furans and certain organics. The EPA has set a limit on these pollutants -- these limits have been significantly reduced over the past 15 years.
Certain hazardous streams will continue to make their way through screening -- and therefore get incinerated. If these hazardous sources are incinerated, the resultant emissions will go into the air.
Once a waste stream has been incinerated, the pollutants will go into the atmosphere and pollutants will not be confined to a known area (compared to the known physical bounds of a landfill).
Traffic around a RRF is significantly increased due to the heavy truck and waste hauler traffic.
There are public perception problems with any facility that has a smoke stack. Neighbors don't enjoy seeing a tall stack looming on the horizon.