A normal direct-drive drill/screw driver which can drive a lag screw into a stud would break your wrist in the process, if you could even hold onto it. The impact driver uses a spring loaded hammer and ratcheting mechanism to apply torque in small increments so as not to rip the tool out of your hand and send it flying.
Generally, it's better to have an impact driver than a drill if you have to pick one. They work just fine for drilling and screwing, you just have to be careful about stripping thing. The only time I use the regular (corded) drill is for hole saws or larger spade bits, because the drill has less play and cuts smoother in those cases.
I would suggest getting a nice impact driver, but if you can't right now, you can also turn the lag bolt manually using a socket and long ratchet or breaker bar.