Different results when using brush blending mode vs layer blending mode

This should work to get the same results on a blank layer:

Layer>New>Layer set Mode to Color Dodge and check Fill with Color Dodge Neutral Color (black).

Set the brush mode to normal and then paint on that layer.

or

Make a new blank layer, set the Blend Mode to Color Dodge, go to Blending Options in the Layer Styles dialog and uncheck Transparency Shapes Layer

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Why the differences seems to be a story in itself.

Photoshop 4 thru 5.5 used Color Dodge and Color Burn the same on transparent layers as painting on a layer with the brush mode set to Color Dodge or Color Burn.

Then in photoshop 6 there were changes to the way Difference, Color Dodge and Color Burn layer blend modes were applied to layers, which was different than the previous way apparently, so thus the differences you noticed between brush color dodge and layer color dodge and the workarounds listed above to mimic the old way.

Here's an old post on it that has a link to a now defunct KB about it.

Photoshop 6 and the change in Color Dodge/Burn blend modes

Here's the KB if you want to read it:

How are the other brush setting set.  Brush paint often has a build up effect and and feathering and other brush option like sampled brush tips etc. Painting on the actual layer changer to pixels you painting on  and  may vary with brush spacing.  Painting on seperated layer the way the blending is done may look a little different for the underlying layers pixels are not changed which may effect how blending looks the paint is not actually blended into the lower layer the fresh paint is on a layer of its own above it.. All that I wrote though is just an educated guess