ASME and IMechE are both very old and prestigious engineering societies. As to their differences, please, follow the links below for a fair description of them: if you read both the Wikipedia texts, and their relevant References, a sound judgement may be made concerning your interests. Also, you may use the provided links in there to get to their home websites, for further information.
In regard to which one is “better”, it all depends on what one is looking for: then, nobody better than yourself for deciding. However, the following procedure may help you to decide: thus, get to the “Journal Rankings” web page of the bibliometric indicators database SJR, by following the next link:
http://www.scimagojr.com/journalrank.php
Now, enter into their Search box ASME to get:
http://www.scimagojr.com/journalsearch.php?q=ASME
showing 234 journals, conferences, etc. published by/related to ASME. Moreover, using SJR webpage reading tools, one may obtain valuable bibliometric indicators for each record, on which to base further decisions. If found useful, please, repeat the procedure with IMechE.
As a experience of being the student member of the both society's , the ImechE focus on student development and also does not charge for membership ,all events organized by them.
when coming to ASME they do not provide the free of charge entry for the all the events.But when compared the ASME as more registered members than IMechE (International)
As a chartered engineer (CEng) registered with IMechE, I would say for a professional engineer, IMechE has more prestige. Getting elected as a chartered engineer or fellow is not easy, i.e. not merely paying a requisite fee. There are multiple evaluations of forms and interviews. However, IMechE is more UK centric just as ASME is more US centric, but international membership outreach of both societies is almost equal. For students, I feel IMechE is again more rewarding in terms of competitions and support they provide.