I can't really answer your question definitively being neither a deep expert in stoicism nor in Peterson's thought (I've done one graduate course on hellenistic philosophy and I've seen some of Peterson's youtube stuff), but given those caveats I will give it a shot.
I would suggest that the two aren't that similar.
First, it helps to understand what stoicism believes and not merely how people imagine stoics to be (perhaps there's quite a bit of slippage here between "stoicism" and the historical philosophical body of thought Stoicism). It's not a philosophy of self-help per se but rather a theory about the nature of the universe, determinism, and our place in it. Specifically, it holds that all things are causally determined and unchangeable -- the only things we can control our responses to the world we are in. Stoicism is also a theory about happiness -- specifically, it maintains that happiness is the use of reason. i.e. to respond to all things virtuously and with ataraxia (without worry).
Differing from Aristotle, the stoics (see for instance Marcus Aurelius) believed that external accoutrements are not necessary for happiness -- only virtue is. And only our vices causes us harm.
Stoicism was one of several competing "philosophies" where philosophy here differs somewhat from the contemporary style and is a type of "life philosophy" and set of religious (or quasi-religious) practices.
In contrast, Peterson appears to be advocating something built on Jungian archetypes and contemporary psychological studies and methods. The Jungian bit is that there's an arc of experiences you need to undergo in order to achieve a good life or happiness (struggle with darkness and overcome or something like that) plus data-based claims about how people should organize their life for good outcomes.
In this sense, it's also a type of life philosophy (or as one commenter derisively puts its "self-help.")
I'm sure there's some superficial overlap in that nearly all life-philosophies would give some of the same advice. E.g., "do something meaningful" is advice that just about anyone would agree to -- until we decide what "meaningful" means.
tl;dr - both are approaches and theories about what you need to live a good life. Beyond that, they don't seem to have much in common.