Do you move in 5' increments on a battle map, or can you move less than that? If so, how do you track that on the map?
If there is, does it create any cases that may be a concern?
Under the Variant: Playing on a grid (PHB, 192) it states, emphasis mine:
Rather than moving foot by foot, move square by square on the grid. This means you use your speed in 5-foot segments. This is particularly easy if you translate your speed into squares by dividing the speed by 5. For example, a speed of 30 feet translates into a speed of 6 squares.
Under the grid system, you move in 5 foot segments, but under Theatre of the Mind, you can move in 1 foot increments. The cases are likely edge cases when this will matter, but it is something to consider and understand when answering questions here regarding movement as well as at your tables.
No, they should be roughly the same.
The grid just makes it more obvious which spaces are controlled, while the theatre of the mind allows for more leeway (through uncertainty). Your position in the grid is more of a representation of your location, rather than your exact position.
The 10' grid space is 6'+ away from the target. Your character is a 5' block (so you occupy the 6'-10' spaces), so most people move in 5' increments (and some DMs might require it), which is for the sake of simplicity if nothing else.
However, with a sufficient grid it could be easy to represent 1' increments. For example, a 1" grid is roughly 2.5sq cm. This breaks down to 5mm sq spaces to represent 1' each. A standard Mini might have a 25mm sq base which perfectly fits a 5' representation on this grid. In this scenario there would be no reason to disallow 1' movements.
Without this style of grid it is simply more difficult to track where your controlled space actually is (60% of this square and 40% of the previous, etc.) but its still very possible. The only thing stopping you from straddling a grid line is a strict DM.