I always thought that these two terms are some kind of synonyms, meaning that if you have a self-similar or scale invariant system, you can zoom in or out as you like and you will always see the same picture (physics).
But now I have just read in the context of a lattice of spins model for example, that if the system is at its critical point and therefore scale invariant, it does not mean that it is self-similar as naively discribed in the first paragraph of this question. The author of the paper I am reading even calls it a false picture on p. 9. Later on on p. 24 he explains that poles on the positive real axis on the so called Borel plane break self-similarity because lead to tha fact that to obtain the effective action (for a $\lambda\phi^4$ bare action) the scale dependent non perturbative power correction terms have to be kept. So if scale invariance and self-similarity are not exactly the same, the explanation for breaking of scale invariance should be (slightly and subtly?) different?