Main Difference
The main difference between Quantile and Quartile is that the Quantile is a cutpoint dividing a set of observations into equal sized groups and Quartile is a the three points that divide the data set into four equal groups in descriptive statistics

Quantile
In statistics and probability quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one less quantile than the number of groups created. Thus quartiles are the three cut points that will divide a dataset into four equalsized groups. Common quantiles have special names: for instance quartile, decile (creating 10 groups: see below for more). The groups created are termed halves, thirds, quarters, etc., though sometimes the terms for the quantile are used for the groups created, rather than for the cut points.
qquantiles are values that partition a finite set of values into q subsets of (nearly) equal sizes. There are q − 1 of the qquantiles, one for each integer k satisfying 0

Quartile
A quartile is a type of quantile. The first quartile (Q1) is defined as the middle number between the smallest number and the median of the data set. The second quartile (Q2) is the median of the data. The third quartile (Q3) is the middle value between the median and the highest value of the data set.
In applications of statistics such as epidemiology, sociology and finance, the quartiles of a ranked set of data values are the four subsets whose boundaries are the three quartile points. Thus an individual item might be described as being “on the upper quartile”.

Quantile (noun)
One of the class of values of a variate which divides the members of a batch or sample into equalsized subgroups of adjacent values or a probability distribution into distributions of equal probability.

Quartile (noun)
Any of the three points that divide an ordered distribution into four parts, each containing a quarter of the population.

Quartile (noun)
Any one of the four groups so divided.
“This school is ranked in the first quartile.”

Quantile (noun)
each of any set of values of a variate which divide a frequency distribution into equal groups, each containing the same fraction of the total population.

Quantile (noun)
any of the groups so produced, e.g. a quartile or percentile.

Quartile (noun)
each of four equal groups into which a population can be divided according to the distribution of values of a particular variable
“in the highest quartile, the mean age was 72”

Quartile (noun)
each of the three values of the random variable which divide a population into quartiles.