it's been 10 days joining stack exchange ! I observed that there are two Astronomy SE, Meta and Beta ? So what is the difference between them ?
The meta site is for asking questions or commenting on Astronomy SE's policies and procedures - actually, just like the question you have asked, congratulations!
The "beta" site is the main site where you actually ask questions about astronomy and astrophysics - check that your question is "on-topic" here.
Just have a look at the first 10 active questions on each and you will get the idea.
NB: Upvotes and downvotes have different meanings on the main (beta) and meta sites. A downvote/upvote on the meta may just mean someone disagrees/agrees with your opinion; whereas you shouldn't ask questions that may be matters of opinion on the main site.
We'll normally call the two sites "the main site" and "meta", or "Astronomy SE" and "Astronomy Meta" or similar.
"Beta" is an adjective, used to describe any Stack Exchange site that hasn't "graduated" yet.
What graduating means, and its importance has changed over the years from how it was originally imagined.
There are many small differences between graduated and un-graduated sites, but to most users there are few noticeable differences beyond the reputation (points) threshold values for various privileges are different between the two kinds of SE sites.
Note there are almost 200 SE sites
Note also that there isn't really a main Stack Exchange site to post questions, but there is the "main meta" site https://meta.stackexchange.com for questions that apply to the whole Stack Exchange network.
By far the most important page in the main meta is the FAQ page.
You can read about the reputation threshold differences between beta and graduated sites in the answer to the FAQ
Curiously though there are no FAQ that contain the word "graduation" or "beta". Hmm...
Your original question (differentiating 'beta' and 'meta') has been aptly answered by @ProfRob and @uhoh. But there is also another angle of your question: what is the difference between 'beta' and 'main SE site' and when does 'beta' become an official SE site?
Let's imagine you are developing and launching a mobile app. Your initial updates will be 'beta' as the updates may come as incomplete/unstable or have limited access. Once the development is good enough and more and more people are using this app, you release the main update. Similar goes for SE sites. When a SE site is proposed in Area 51, it subsequently goes to 'beta' period where it has somewhat limited privileges and access. It has beta reputation threshold privileges, doesn't have customized design, doesn't have community adds, doesn't have migration path etc.
You must be thinking: "this site has healthy participation, healthy number of question and answers, so what is preventing it from becoming a main site?" Actually, all those metrics doesn't matter. Our goal is to keep the site healthy and stable. Eventually, the Stack Exchange staffs will deem the site to be ready for graduation (and no, moderators don't have the power the graduate the site on their own).
There were few discussion that happened before. You can go through them to have a better idea: