So, how does it compare to MathJobs? Obviously one comparison is the price point. Both sites have two types of listings: advertising only, and a application management system. For MathJobs, the current pricing is, quoted from their page:
Regular Account (for up to seven ads, full functioning, 12 months from date of sign-up), US610
Regular Account (for one ad only, full functioning, 12 months from date of sign-up), US415
Advertising-only (for up to seven ads, no online applications, 12 months from date of sign-up), US495
Advertising-only (for one ad only, no online applications, 12 months from date of sign-up), US305
Upgrade from existing single-ad account to seven ad account, US295
For MathHire (for academic jobs), the 'advertising only option is free, and the advertising and application management system is supposed to be 200 euros (currently about 220USD), but all jobs posted before April 1, 2016 will be free as well. This makes sense — the system is new and it needs users to be sustainable.
MathJobs, despite its antiquated 90's look, is definitely the more polished looking of the two in terms of usability. For example, the UI of MathJobs is much better at presenting information in a succint way. The MathHire website does not take up the full browser screen width and requires lots of scrolling to see a small amount of information. Also, it was difficult to tell from the job listings page which ones had the feature for applying on MathHire, whereas on MathJobs this is evident. The top bar on MathHire takes up too much screen space, especially on small-screened laptops.
However, I admire MathHire and I think it's an interesting creation that deserves attention. For one, as we've seen, posting only is free. Given the extremely low technical requirements for managing a posting-only system, it seems a bit exorbitant to charge 305 dollars for one ad, as MathJobs does. MathHire also promises more features for reviewing applications from their website. I can't comment too much on this, since I've not reviewed jobs on either sites.
If MathHire would work on making an intuitve, polished UI (which is one of the most important factors for the success of an online community, cf. stackexchange), I believe it would have a chance to overtake MathJobs eventually.
Update: since this post, the MathHire website user interface has been updated in response to some of the comments in this post and looks good. I encourage you to check it out!