At the risk of oversimplifying: In order to have DEEP bass, a woofer needs to have a low free-air resonance frequency; a motor that is neither too strong nor too weak for the type of enclosure used (usually it's a vented box); and it needs to either be in a large enough enclosure to enable deep bass response, OR it needs to use EQ to get deeper bass extension than would otherwise have been possible.
In order to have LOUD deep bass, a woofer needs to move a lot of air.
The amount of air a woofer can move is its surface area multiplied by how far the cone can travel. So it is possible for a given 6" woofer to move more air than a given 8" woofer; it just has to move far enough to make up for its smaller cone area.
In general, the parameters that are conducive to deeper and louder bass are more likely to be found in an 8" woofer than in a 6" woofer, but the enclosure gets a vote too: A "typical" 8" woofer in a small enclosure will probably not go as deep as a "typical" 6" woofer in a larger enclosure.
Also be aware that designers often choose to use woofers with "atypical" characteristics when they're trying to do things like get very deep bass out of a fairly small box. In this situation, the designer would trade off efficiency in order to get deeper bass in a small box.
Much of what I've said here is generalization, and the specifics of the individual designs matter more than generalizations.
Duke

