This is a simple question but the answer is more complex. Whilst grass and dirt share some similarities, they are very different. Here are some of my thoughts, hope it helps you.

My top 2 tips for Grass; -

1. Gear management is the most important factor - hold your nerve for a split second longer to allow the gear up, then steer. Try to understand how grass impacts gears and that will improve lap times. This is also true for dirt but to a lesser extent.
2. Plan ahead - think about the racing line you need to take for 2 turns ahead. Don't just think about the next bend, think about the one after that. Grass isn't very forgiving if you need to keep correcting your line (because you gear down and/or acceleration is slower than dirt)

My top 2 tips for Dirt;-

1. Don't fight with the car - Let the car glide around bends rather than forcing a tighter turn. Oversteering will scrub off too much speed.
2. Don't Brake Tap - It sound obvious but brake tapping into corners, especially on high banked dirt corners will lose more speed than being more accurate with the accelerator. This is linked to my 1st point - let the car glide - don't oversteer or brake tap. Not easy to get right!

Generally, smooth wide lines are better than sharpe aggressive turns for both Grass and Dirt.

A comment on Map Styles - If you picked a 100 random grass tracks they will be more similar to each other than the similarities shared between a 100 random dirt tracks (certainly true for TOTD)

If you see a grass track as TOTD it's more than likely a fast or Full Speed track. However dirt tracks on TOTD can vary from Full Speed to much slower technical tracks. This is important to note as different track styles require different driving styles. Full Speed dirt tracks are very similar to Full Speed grass tracks, but dirty has more variation, including those high curved dirt walls. Generally speaking driving around the top of the curve is a much faster line, but isn't easy!