Main Difference
The main difference between Variance and Standard Deviation is that the Variance is a expectation of the squared deviation of a random variable from its mean and Standard Deviation is a dispersion of the values of a random variable around its expected value

Variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean. Informally, it measures how far a set of (random) numbers are spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by
σ
2
{displaystyle sigma ^{2}}
,
s
2
{displaystyle s^{2}}
, or
Var
(
X
)
{displaystyle operatorname {Var} (X)}
.

Standard Deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.
Standard deviation may be abbreviated SD, and is most commonly represented in mathematical texts and equations by the lower case Greek letter sigma σ, for the population standard deviation, or the Latin letter s, for the sample standard deviation.The standard deviation of a random variable, sample, statistical population, data set, or probability distribution is the square root of its variance. It is algebraically simpler, though in practice, less robust than the average absolute deviation. A useful property of the standard deviation is that unlike the variance, it is expressed in the same unit as the data.
The standard deviation of a population or sample and the standard error of a statistic (e.g., of the sample mean) are quite different, but related. The sample mean’s standard error is the standard deviation of the set of means that would be found by drawing an infinite number of repeated samples from the population and computing a mean for each sample. The mean’s standard error turns out to equal the population standard deviation divided by the square root of the sample size, and is estimated by using the sample standard deviation divided by the square root of the sample size. For example, a poll’s standard error (what is reported as the margin of error of the poll), is the expected standard deviation of the estimated mean if the same poll were to be conducted multiple times. Thus, the standard error estimates the standard deviation of an estimate, which itself measures how much the estimate depends on the particular sample that was taken from the population.
In science, it is common to report both the standard deviation of the data (as a summary statistic) and the standard error of the estimate (as a measure of potential error in the findings). By convention, only effects more than two standard errors away from a null expectation are considered “statistically significant”, a safeguard against spurious conclusion that are really due to random sampling error.
When only a sample of data from a population is available, the term standard deviation of the sample or sample standard deviation can refer to either the abovementioned quantity as applied to those data, or to a modified quantity that is an unbiased estimate of the population standard deviation (the standard deviation of the entire population).

Variance (noun)
The act of varying or the state of being variable

Variance (noun)
A difference between what is expected and what happens

Variance (noun)
The state of differing or being in conflict

Variance (noun)
A discrepancy, especially between two legal documents

Variance (noun)
The second central moment in probability

Variance (noun)
covariance and contravariance generally

Variance (noun)
the fact or quality of being different, divergent, or inconsistent
“her light tone was at variance with her sudden trembling”
“the stylistic variances of classical dance”

Variance (noun)
the state or fact of disagreeing or quarrelling
“they were at variance with all their previous allies”

Variance (noun)
a discrepancy between two statements or documents.

Variance (noun)
a quantity equal to the square of the standard deviation.

Variance (noun)
an official dispensation from a rule or regulation, typically a building regulation.