Main Difference
The main difference between Average and Median is that the Average is a colloquial term for the central tendency (middle or typical number of a list of numbers, including mean, median and mode) and Median is a quantile

Average
In colloquial language, an average is a single number taken as representative of a list of numbers. Different concepts of average are used in different contexts. Often “average” refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged. In statistics, mean, median, and mode are all known as measures of central tendency, and in colloquial usage any of these might be called an average value.

Median
The median is the value separating the higher half from the lower half of a data sample (a population or a probability distribution). For a data set, it may be thought of as the “middle” value. For example, in the data set {1, 3, 3, 6, 7, 8, 9}, the median is 6, the fourth largest, and also the fourth smallest, number in the sample. For a continuous probability distribution, the median is the value such that a number is equally likely to fall above or below it.
The median is a commonly used measure of the properties of a data set in statistics and probability theory. The basic advantage of the median in describing data compared to the mean (often simply described as the “average”) is that it is not skewed so much by a small proportion of extremely large or small values, and so it may give a better idea of a “typical” value. For example, in understanding statistics like household income or assets, which vary greatly, the mean may be skewed by a small number of extremely high or low values. Median income, for example, may be a better way to suggest what a “typical” income is.
Because of this, the median is of central importance in robust statistics, as it is the most resistant statistic, having a breakdown point of 50%: so long as no more than half the data are contaminated, the median will not give an arbitrarily large or small result.

Average (noun)
The arithmetic mean.
“The average of 10, 20 and 24 is (10 + 20 + 24)/3 = 18.”

Average (noun)
Any measure of central tendency, especially any mean, the median, or the mode. from c. 1735

Average (noun)
Financial loss due to damage to transported goods; compensation for damage or loss. from 15th c.

Average (noun)
Customs duty or similar charge payable on transported goods.

Average (noun)
Proportional or equitable distribution of financial expense.

Average (noun)
An scoring record, etc.
“batting average”

Average (noun)
In the corn trade, the medial price of the several kinds of grain in the principal corn markets.

Average (noun)
The service that a tenant owed his lord, to be done by the animals of the tenant, such as the transportation of wheat, turf, etc.

Average (adjective)
Constituting or relating to the average.
“The average age of the participants was 18.5.”

Average (adjective)
Neither very good nor very bad; rated somewhere in the middle of all others in the same category.
“I soon found I was only an average chess player.”

Average (adjective)
Typical.
“The average family will not need the more expensive features of this product.”

Average (adjective)
Not outstanding, not good, banal; bad or poor.

Average (verb)
To compute the average of, especially the arithmetic mean.
“If you average 10, 20 and 24, you get 18.”

Average (verb)
Over a period of time or across members of a population, to have or generate a mean value of.
“The daily high temperature last month averaged 15°C.”
“I averaged 75% in my examinations this year.”

Average (verb)
To divide among a number, according to a given proportion.
“to average a loss”

Average (verb)
To be, generally or on average.

Median (noun)
A central vein or nerve, especially the median vein or median nerve running through the forearm and arm. from 15th c.

Median (noun)
A line segment joining the vertex of triangle to the midpoint of the opposing side.

Median (noun)
The number separating the higher half of a data mean of the two middle values. from 19th c.

Median (noun)
The median strip; the area separating two lanes of oppositedirection traffic. from 20th c.

Median (adjective)
Situated in the middle; central, intermediate. from 16th c.

Median (adjective)
In the middle of an organ, structure etc.; towards the median plane of an organ or limb. from 16th c.

Median (adjective)
Having the median as its value. from 19th c.

Average (noun)
a number expressing the central or typical value in a set of data, in particular the mode, median, or (most commonly) the mean, which is calculated by dividing the sum of the values in the set by their number
“the proportion of over60s is above the EU average of 19 per cent”

Average (noun)
an amount, standard, level, or rate regarded as usual or ordinary
“underground water reserves are below average”
“they take about thirty minutes on average”

Average (noun)
the apportionment of financial liability resulting from loss of or damage to a ship or its cargo.

Average (noun)
reduction in the amount payable under an insurance policy, e.g. in respect of partial loss.

Average (adjective)
constituting the result obtained by adding together several amounts and then dividing this total by the number of amounts
“the average temperature in May was 4°C below normal”

Average (adjective)
of the usual or ordinary amount, standard, level, or rate
“a woman of average height”

Average (adjective)
having qualities that are seen as typical of a particular person, group, or thing
“the average lad likes a good night out”

Average (adjective)
mediocre; not very good
“a very average director making very average movies”

Average (verb)
amount to or achieve as an average rate or amount over a period of time; mean
“annual inflation averaged 2.4 per cent”

Average (verb)
calculate or estimate the average of
“they earned only £35 weekly when their seasonal earnings were averaged out”

Average (verb)
result in an even distribution; even out
“it is reasonable to hope that the results will average out”

Average (verb)
result in an average figure of
“the cost should average out at about £6 per page”