Age spots and liver spots are the same.
More Info: Age spots and liver spots are common alternative names for the medical condition known as senile lentigo. 
Liver spots and age spots are not the only alternative names for senile lentigo. One quick internet search will result in a lengthy list of variations including solar lentigines, solar lentigo, senile lentigines, senile lentigo, sun spots, lentigo senilis, and senile freckle to name a few.
What’s In a Name?
Understanding what they are will help to clarify why they go by so many different names. Senile lentigo is defined as: “a benign pigmented flat spot on sun-exposed skin in older adults, especially on the back of the hands and on the forehead. Also called a liver spot.”  Considering singular definitions clarifies why they are so named.
Lentigo: a type of freckle that is a small tan, brown, or black spot, which tends to be darker than the usual.
Lentigines: plural form of lentigo.
Senile: Of or relating to old age.
Solar: Relating to or caused by the sun.
Why Are Age Spots Called Liver Spots?
The fact that 90% of Caucasian Americans over the age of 60 have senile lentigines clarifies why so many people refer to them as ‘age spots’.  But why are they sometimes referred to as ‘liver spots’? In the past, it was believed that these patches of skin discoloration signified a problem with the liver. Medical science has since proven that theory to be incorrect, but there are several holistic healing books that continue to promote the fact that liver spots are an indication that the liver is congested and needs to be cleansed. 
You Can Prolong Their Appearance
Whichever name you choose to call them, there are measures that you can take to prolong their appearance. Years of sun-exposure is the primary culprit in the appearance of age spots. Take measures to protect yourself from the sun as early in life as possible by wearing protective clothing and sunglasses, wearing a quality sunscreen, and avoiding sun exposure at midday when it is the most intense. 
Definition of Liver Spot
Senile Lentigo Definition
 Medscape Reference
 University of Maryland Medical Center
 Board, A.D.A.M