What is the difference between the parameters and the moments of a distribution? What is the difference between the parameters and the moments of a distribution? Are the moments of the distribution (mean, sd etc.) simply also parameters of a distribution?
 A: The moments of a distribution are defined as expected values of specific functions of the random variable following this distribution. Raw moments $\mu'_k \equiv E(X^k)$. Central moments: $\mu_k \equiv  E[X- E(X)]^k$. Fundamentally $k$ is taken to be an integer, but there is also theory developed for fractional moments (i.e. with $k$ not being an integer). 
The parameters of a distribution appear in and characterize the distribution (CDF) and density (PDF) functions of a distribution. Since in order to calculate an expected value under a distribution we will use its density function, it should come as no surprise that, if we actually perform the integration implied by the expected value operator, the moments will eventually be expressed as functions of these parameters.  
For example for a continuous uniform random variable $X \sim U(a,b)$, the parameters are the lower and upper bound of the support of the random variable. The distribution has density function $f_U (u) = 1/(b-a)$. Then the first raw moment is the mean and 
$$\mu = E(X) = \frac {a+b}{2}$$
(note the established notational inconsistency here: while in general raw moments are denoted with a prime, the mean especially is written without one)
while the second centered moment is the variance and 
$$\mu_2 = \text{Var} (X) =  E[X-E(X)]^2 = \frac {(b-a)^2}{12}$$
In some special cases, the normal distribution being the most well-known and predominant one (and perhaps the source of OP confusion), some basic moments of the distribution are equal to a given parameter of the distribution, and not to a function of all of them. The normal distribution has parameters $(\mu, \sigma^2)$, and we find out that $\mu = E(X)$ and $\sigma^2 = \mu_2$ (see this answer for the proof). But in general, each moment is a function of all the parameters characterizing a distribution.
