The amount of time after a domain expires before you can register it varies from company to company. They decide how long they hold on to it before they release it back to the "pool" of availble domains.
In the case of Network Solutions, I've seen domains that had been expired for TEN MONTHS but still not available for purchase. So, it could be a long time.
In theory, yes, companies like SnapNames are more likely to be able to register the domain before you can. The reason is that they can watch the domain 24 hours a day if they want. You, as a human being, just can't match that. Theoretically, they could have it registered 60 seconds after it's released, where you might not even know that it's available for hours or even days after it happens.
If it's is REALLY important that you get a particular domain, spend the $$$. Take it from me -- I got burned by not doing so.