what is the difference between mixture of two normal distributions and sum of two independent variables The following denotes a mixture of a standard normal with a
normal with the same mean but 100 times the variance:
$0.95 \mathrm{~N}(0,1)+ 0.05 \mathrm{~N}(0,100)$
Let Y = 0.95 X + 0.05 Z with X,Z are independent normal distribution with X ~ N(0,1) Z~N(0,100)
I remember Y = X + Z if X ~ N(u1,s1) and Z ~ N(u2,s2), then Y would follow N(u1 + u2,s1+s2)
These two conclusions seems different and makes me confused.
Hope someone can explain

Another way to think about it: if $X$ is generated from one distribution and $Y$ from another, then $X+Y$ is a sample from the distribution of the sum.  To get a sample from the mixture, you pick either $X$ or $Y$, with probabilities given by the mixing proportions.
The distribution of the sum is 'more Normal' than either of the component distributions: adding two distributions tends to smooth out their special features.  The distribution of the mixture isn't; a mixture of two Normals is bimodal, because you get an observation from one or the other.
