When I read it correctly, the question was about the conversion between chlorophyll per volume and chlorophyll per area. This question is indeed not uncommon for comparing different aquatic systems with different depth, especially as macrophytes grow per area and also light is measured "per area". On the other hand, plankton grows (and is measured) per volume. Therefore, conversion between 1/m3 and 1/m2 needs an additional information: water depth. The per area chlorophyll is simply the integral (or sum) over all depths.
Dear Julie, one additional note about sampling. Just a single value from the water surface will not be representative for deep stratified lakes. Recommended is either an integrated sample (representatively mixed over the depth range) or a depth profile, from which you can calculate a weighted average resp. integral. R package marelac has a function for depth or volume weighted averaging.
The two are not the same. As Thomas Petzoldt noted above, the first is per volume (1 liter = cubic meter) and the second is per area (square meter). How to compare the two would depend on what the chlorophyll values are from (e.g., a leaf, periphyton, or phytoplankton).