Maximum and maximal take things to the max. In everyday speech and writing, both words describe the largest, greatest, fastest, or best of something. But while maximum can be an adjective or a noun, maximal should only be used as an adjective.
Maximum is "greatest" in Latin, from the root magnus, "great or big." Whether it's being used as a noun or adjective, maximum means the upper limit, or the most that's possible or allowed:
Coaches say the expansion from a maximum of seven days to a maximum of nine days has made for a better tournament. (The Seattle Times)
In order to get the maximum number of presents, Sally had invited every single person in our class. (The Lions of Little Rock)
Maximal derives from the same Latin root as maximum and means "the highest or most comprehensive." It's often found in technical or scientific contexts and is commonly used to contrast with its antonym, "minimal:"
According to the scientists' calculations, his average pace represented about 85 percent of his maximal endurance capacity, meaning he was pushing himself hard. (New York Times)
A ban on plastic and a charge for paper has the potential to achieve maximal environment gain with minimal business disruption. (San Diego Union-Tribune)
These two words can often be interchangeable when used as adjectives. For example, using maximum force or maximal force to pry something open are both valid. But when in doubt, go with maximum for everyday contexts and use maximal in more technical contexts.
Another area where these two are confused are in mathematical and computer science writing. In this case, the subtle difference between the two is that maximum describes the largest value, but maximal is used for the largest element in a specific category or group. For example, the maximum of the numbers 2, 5, and 7 is 7. Among the numbers 1–7, the maximal set of odd numbers has four elements: 1, 3, 5, 7 — because you can’t add any more without breaking the rule.