This is the most common confusion on the shelf and it has a simple answer: a dobsonian is a newtonian. The two words describe different parts of the same instrument, and once that is clear the real question becomes which mount suits you.
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What each word describes
Newtonian is the optical design: a curved primary mirror at the bottom of a tube and a small flat secondary near the top that sends the light out the side to an eyepiece. Dobsonian is the mount: a simple box that turns and tips. So every dobsonian is a newtonian, and a newtonian on an equatorial mount is a newtonian that is not a dobsonian.
What the mount changes
An equatorial mount has one axis aligned with the Earth's rotation, so a single slow motion, or a motor, follows an object across the sky. That is essential for photography and pleasant at high magnification. It also costs money, adds weight, needs setting up and, on cheap examples, wobbles. A dobsonian base is steadier per pound than almost any tripod.
Newtonian vs dobsonian telescope: which to buy
For looking, a dobsonian, because the same money buys more mirror. For photographing, a newtonian on a driven equatorial mount, accepting a smaller aperture for the same price. Either way, newtonian reflector vs dobsonian is a choice of mount: the reflector itself is identical in both cases and so is the view it gives.
Questions people ask about dobsonian
Is a dobsonian reflector a newtonian telescope?
Yes. Dobsonian refers to the mount, newtonian to the optics. A dobsonian is a newtonian reflector on a simple alt-azimuth base.
Which is better, a dobsonian or a newtonian on an equatorial mount?
For visual use the dobsonian, because more of the price is in the mirror. For astrophotography the equatorial, because tracking is not optional there.
Can you put a dobsonian tube on an equatorial mount?
Yes, and people do. The optical tube is a standard newtonian, so it moves between mounts with the right rings or dovetail.