Is Science Publishing group (http://www.sciencepg.net) a genuine publisher related to the Science journal? I received an email requesting me to apply for a reviewer/editor position at Science Publishing group. But I found that Science Publications have a different website, hence I feel that the email is a fraud. Please help to clarify matters.
Science is the official magazine of the American Association for the Advancement of Science, which has nothing to do with two publishers you mentioned. It is common these days that new publishers (usually publishing online open access journals) use the names, which are similar to the famous publications.

Science and Nature are the most reputable journals (though they are technically magazines) in cutting-edge research. Both have impact factors over 30. The publishers you mentioned publish journals, which are not even properly indexed. Most of the journals by such publishers have an impact factor of around 1 (just mentioned to highlight the difference).

In any case, no reputable publisher asks you to apply. They choose their editors/reviewers directly based on your works. When an email asks you to apply, this is a general advertisement email sent to a long mailing list.