The flat plane and height don't really have anything to do with whether the datum is important. It just adds another variable.
A "datum" is a type. So, let's say you put something on a map and used a ruler. You could say the location is one inch from the top margin and two inches from the right margin. Someone else might print the map on a different sized piece of paper and say the location is 1.5" from the top and 2.25" from the left margin. That's not what datum means specific to lat/lon, but what it is is more the way you measured the earth would put lat/lon in different places...man I don't know how to explain things to a logger

It's two ways of giving the coordinates and they don't necessarily lay at the "exact" same spot. It's not metric, but I'll use that as an example...say you're going 55, well, is it mph or kph (kilometers).
Here's wiki:
"Latitude and longitude values can be based on several different geodetic systems or datums, the most common being WGS 84 used by all GPS equipment.[5] Other datums however are significant because they were chosen by a national cartographical organisation as the best method for representing their region, and these are the datums used on printed maps. Using the latitude and longitude found on a map may not give the same reference as on a GPS receiver. Coordinates from the mapping system can sometimes be changed into another datum using a simple translation. For example, to convert from ETRF89 (GPS) to the Irish Grid add 49 metres to the east, and subtract 23.4 metres from the north.[6] More generally one datum is changed into any other datum using a process called Helmert transformations. This involves converting the spherical coordinates into Cartesian coordinates and applying a seven parameter transformation (translation, three-dimensional rotation), and converting back.[1]
In popular GIS software, data projected in latitude/longitude is often represented as a 'Geographic Coordinate System'. For example, data in latitude/longitude if the datum is the North American Datum of 1983 is denoted by 'GCS North American 1983'"
Most USGS maps are not in WGS84, but your GPS is. If you get a lat/lon on your GPS then try to find it on a USGS topo map it'll be off by a little, same general area, but not the same specific area. Most USGS topo maps are a NAD datum (North American Datum, 27 or 83, indicating the year).
If you have a GPS, look at the datum. You should be able to change it. If you change it from WGS84 to something else your waypoints will have a different location. Write down the lat/lon of a waypoint and then change the datum used by the GPS. They should change. It's important to know your datum, really important actually. If you have a ground crew looking for you using USGS topo maps and you're using your GPS to give your location they'll likely be looking for you in the wrong place. Everyone has to use the same datum to be able to find the same place, even with the same lat/lon coordinates.
'nother example. Say you and I each go to a place on the earth and take a picture of the tree line, but I'm standing 1/4 mile farther to the west than you. If I say, hey look what's in the middle of my picture it won't be what's in the middle of your picture. It'll probably be in your picture, but it won't be the center of the picture. Datums are different views of the earth when making a map or referencing a place on the earth, it shifts the locations depending on which view (datum) you're using.