Hi there,
Do you believe that having an old domain can help you to get better ranking for your desired keywords?
So why some of new domains can compete with old ones in some cases?
Any ideas?

No. I think age in itself does not guarantee ranking. Unless an aged domain enjoys good reputation and quality backlinks, it won't help you much. And that's exactly the reason why new domains can compete with old ones.

As we all know,if we have new domain so we are afraid to promote it heavily(we think,Google could be make it in "Sandbox). Domain age does matter in terms of SEO.

In terms of SEO, domain age matters in terms of google algorithm but only one point is not capable enough for ranking. You have to strictly follow all SEO guideline which help to kill competition and give you batter ranking in search engine.

Yes i think domain age matters in seo , because old domain can help you in ranking ,for new domain you cant make too many links in short time that's why you will need to wait and make links slowly only in that case old domain is good. Otherwise with both new and old you can get success.