Geoff's answer is relevant. A few more things to add:
There are two things that a bigger aperture will provide: more resolution, and more brightness.
In terms of resolution, things go downhill pretty fast after 150 mm (6"). You'd think a 300 mm (12") aperture scope would provide 2x more resolving power, but in reality it's not that simple. There's something called "seeing", which is the stillness of the air that allows you crisp sharp views, and it varies a lot with the place and time. Realistically, instruments bigger than yours (which has about 0.7 arcsec resolving power) will use their resolving power much less often, only during those fleeting moments of perfect seeing. The bigger the scope, the less often you're able to use its full resolving power, because atmosphere is turbulent and keeps churning all the time.
E.g., a 100 mm (4") instrument will provide 1 arcsec resolving power. How often you'll get 1 arcsec seeing, or better, in the place you live? Not very often, unless you live in California, or Hawaii, or some other place with excellent seeing, far from the jet stream. People living in the american north-east often find that instruments bigger than 4...6" are seeing-limited a lot.
Related to resolution, an alternative way to look at it - magnifications up to 100...150x are basically almost always usable. Around 200x, depends on seeing. Over 300x become less often achievable. Beyond that, it's pretty rare that you could crank up the mags without getting a large blurry blob instead of a crisp image.
Now, the second issue, brightness. A 300 mm scope will capture 4x more light than your 150 mm. So it would allow you to see fainter objects. But remember, if you live in the city, there's something called light pollution, due to all the lights around you, that makes the sky glow with light, so the faint objects are hard to see. You'd have to drive an hour or two to the dark skies away from the city to see all the really faint fuzzies. If you own a huge 24" dob then you could only exploit all its light-gathering power in those rare places with zero light pollution.
So, realistically, there are limits to what a big scope could do. Many old experienced users say that a 300 mm (12") f/5 truss dobsonian is the biggest that can be used without major effort. If it's a tube design, instead of truss, then it must be even smaller. Opinions about the real limit will vary between 10" and 16", with most of these scopes being f/4 to f/5.
It's best if you could go to a star party and test a few different scopes yourself.
In any case, the 6" will last you a long time and it's a quite capable instrument. Enjoy.