More links with anchor text and higher PR links?
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What is the domain authority of the competing site compared to yours?
When you say your link profile is better in terms of number and quality, what about trust? Have you compared the mozRank and mozTrust of the 2 sites?
For your link profile, can you give us the number of external followed links, followed links root domains, and unique C-class IP addresses that point to you and them?
What is their domain age compared to yours?
Lastly, if you added all your links recently it might take some time for the rankings to reflect that. Does the rate of link acquisition look normal for you and for them, or have they been adding them steadily over time whereas you added a whole bunch within 2 weeks?
Need a bit more data...
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There are a number of factors to consider.
The links:
(i) Can take time for Google to pick them up. Are they showing in your webmaster tools?
(ii) Do the links have a nofollow tag?
(iii) How competitive is the first page ?
(iv) In one of the latest SEOMoz reports, it has been suggested that exact keyword usage in anchor text may be less effective than broad usage.
(v) How many is double ? from 1 link to 2 ?
(vi) On-page content good? Refreshed it lately ?
It is fine to redirect the non-www to the www with a 301 redirect.
Hope the above helps!