Some sounding rockets are sub-orbital. Sub-orbital merely means going into space with less than orbital velocity. Going into space: That's a bit arbitrary, but a person who goes 50 miles or higher (~80km) gets astronaut wings; another definition is the nice round figure of 100 kilometers (~62 miles) in altitude; that's the Kármán line. Yet another is the nice round figure of 400,000 feet (~122 km; ~76 miles); that's "entry interface" for NASA vehicles that reenter the atmosphere.
By whatever definition one uses, some sounding rocket flights are less than sub-orbital. Sounding rockets sent into the stratosphere may top out at much less than 50 miles altitude. A balloon is cheaper than a rocket, but balloons can only go to an altitude of 31 miles / 50 kilometers or so. If your experiment needs to go higher than that you need to use a sounding rocket.
As far as acceleration is concerned, sounding rockets go whoosh! and they're gone. However, their payload capacities are rather small. They aren't intended for carrying massive people and their even more massive environmental control and life support systems.
See the NASA Sounding Rocket Handbook for more info on sounding rockets and their capabilities.