I'm much older now, and I still have trouble telling them apart...

"facepalm"

Are there any good tips or tricks to help me remember which direction each runs?

Generally I just go with that, it automatically defines the other term... but if you do want to go further with longitude, I agree that using terms with LONG do the job. To get the longitude, you go ALONG the rungs LONGWAYS. The vertical lines are the hashmarks to show how far you've come along.

I'm convinced in my years thinking about this personally and then teaching that a big part of the challenge of any imagery/phrases is the confusion between whether you're talking about the line itself or values. It surely confused me to no end that values of latitude were north/south, while the lines of latitude were drawn east-west. I think/hope the ladder/latitude connection is a clear way to help people of all ages overcome the confusion.

In my mind, the picture of the Earth shows the Earth with the North and South poles at the top and bottom (or reversed, doesn't matter). High describes a vertical orientation. Therefore, the latitude changes in North-South direction.

Additionally, because the horizontal degree scale has a discontinuity at 360 degree (0 degree), it does not make much sense to distinguish high and low values in horizontal direction. Hence, high values are in vertical direction (= latitude).

(The Editors of Encyclopædia Britannica. "Plateosaurus." Encyclopædia Britannica. Encyclopædia Britannica, Inc., 25 May 2010. Web.)

Longitude: A LONG neck dinosaur has a long neck that extend vertically upwards. The lines of longitude extend vertical on a map of the Earth.

Latitude: The back of a long neck dinosaur is relatively FLAT. The lines of latitude lay horizontal or "flat" on a map of the Earth.

TLDR; FLAT back, LONG neck. I hope this helps.