If the brain is not destroyed, the remains of the infected will continue to animate, no matter how dismembered they may become. Crawlers usually don't pose much of a threat, as they often lack mobility.
Crawlers are one of two types of infected that are extremely easy to fight (with the other being distant walkers), but also provide very little xp. Many times, they do little or no damage, and are always immobile so when facing multiple infected, players may want to target them last.