Standardization does not change the skew of the distribution. What it does is transforming the values so it has $\mu=0$ and $\sigma^2=1$. The actual distribution shape remains unchanged.
Log-transformation, on the other hand, changes the skew of the distribution, and is useful when you deal with values that have right-tailed distribution.
Consider the following example.
set.seed(0)
X <- rlnorm(1000)
hist(X)

You can see that X has skewed distribution (long right tail).
Now we do two transformations as follows:
Z <- (X-mean(X))/sd(X)
L <- log(X)
par(mfrow = c(1,2))
hist(Z)
hist(L)

As you can see, standardization of X (which is Z in this example) has similar shape of distribution with X. Log-transformation of X (which is L in this example) is now transformed in a way that eliminates its long right tail.
Note that X in the previous example has lognormal distribution X <- rlnorm(1000)
, that's why the log-transformed variable looks normally distributed. If X is, say, exponentially distributed, the resulting log-transformed variable may not be normal, as shown below.
set.seed(0)
X <- rexp(1000)
par(mfrow = c(1,1))
hist(X)

Z <- (X-mean(X))/sd(X)
L <- log(X)
par(mfrow = c(1,2))
hist(Z)
hist(L)

You can see that the distribution of log-transformed X (denoted by L) now does not have right tail, but still not looking like normal distribution because it's still skewed. Log-transformation does not always make the distribution normal.