The recruiter is pushing because she wants her comission on the books, she'll get her piece once you are in place, pure and simple she wants her money. I know a fair amount of recruiters in So Cal after 30 years in IT. I will take a contract from anyone if I need it but if I'm looking the list is really short, sorry if this offends but 85% of them suck and will be selling something else in 5 years.
DDK632 nailed the types of workers in IT today, be person two if you expect to survive this industry for any length of time in a technical path, if in a management path pick a company and stay there.
Honestly if you accept the offer do not do the counter offer thing you will piss off both your current employer by subjecting them to the pressure, the propsective employer and the recruiter. If you piss off the later two believe people remember and have no problem trashing others if it benefits them.
Good luck move on for the better compensation and make sure yuor skill set gets advanced, and big prominent projects come your way.