Ash is going to raise the pH of the soil. I'm not familiar with the use of biomass briquettes but I could see a negative effect of adding that ash to the soil if the quantity were significant.
Nitrogen is lost (along with other elements) as a result of the burning process, so there's a difference for sure.
A benefit of the compost is that it helps develop a much more complex soil structure, making a sandy soil less sandy and a clay soil less dense. It also provides a much more hospitable environment for microbes and worms.
Ash is going to really just provide a certain set of minerals and elements (calcium carbonate, potash, etc.) without improving the soil structure. I would say it is akin to spreading lime or bagged fertilizer on the ground rather than compost. The former will change the chemical composition but the latter will do that and develop better soil structure.
Your question "what would I lose if I used ash only" is (at least) the following: potential soil building of the compost, the nitrogen that is lost during the burning process, etc. Ash is, however, useful particularly if you are trying to raise the pH of the soil.
From a work-standpoint, well, ash weighs considerably less than compost. It is dried and has released much of its original contents as heat and gas so there's that. It requires less of your energy to spread about, that's for sure.
Me? I'd probably opt to compost it because I'm trying to improve the native clay soil I have here in Virginia. But I'm not strictly opposed to adding ash to the soil. It's a time-tested means of changing the pH and encouraging some plants to grow well.
It isn't, however, an all-or-nothing affair. You could, of course, do both.
One more thing I thought about was weed seeds. If the plant waste you are talking about includes weeds, those would be better as ash unless the compost pile gets hot enough to kill them. Just a thought.