What is the difference between "minimum" and "minimal"?

There's a valid reason for this, which is linked to the fact that both mathematics (as a field) and minimum (as a word) are defined by quantities. Per Oxford Dictionary, minimum: (Note the use of "quantity").

The least or smallest amount or quantity possible, attainable, or required.

As defined by the Oxford Dictionary, minimum is a quantitative representation of the smallest amount needed; thus, making it suitable for math and the term minimum angle.

That brings us to minimal, which can be both a qualitative and quantitative characteristic. Thus, depending on whether the user intends to use it qualitatively or quantitatively he or she may be correct. But regardless of the user' intent, using minimal in this sense is subject to misconception.

Per Merriam Webster...

minimal: barely adequate

Per Merriam Webster: (note the use of "quality")

adequate: good enough : of a quality that is good or acceptable

Therefore, minimal is a qualitative characteristic, which contradicts the quantitative nature of math.

To conclude, I admit that minimal can in some instances be synonymous with minimum; however if you consider the dilemma people will encounter (is minimal qualitative or quantitative?), you'll understand why minimum (only quantitative) better represents mathematics. This explains why many have chosen to use minimal angle rather than minimum angle; however, minimum is more technical.

Minimum is fairly absolute and solid, and refers to the smallest number or amount possible.

Minimum: the least or smallest amount or quantity possible, attainable, or required.

Minimal is a little more flexible, where it refers to being the smallest amount or degree in non-absolute terms.

Minimal: of a minimum amount, quantity, or degree.

"Of a minimum amount", without specifying the minimum amount.

An example of both might be:

Jane has minimal interest in going to the movies; mostly because it seems the ticket prices rise at a minimum of $1.50 every time.

I keep the amount of furniture in my room at a minimum because I like my room to be minimal. I'm a minimalistic person.

According to Dictionary.com

Minimal is an adjective and Minimum is a noun. They have different meanings and purposes.

Minimal

[min-uh-muh l]

adjective

1. constituting a minimum:

a minimal mode of transportation.

2. barely adequate or the least possible:

minimal care.

Minimum

[min-uh-muh m]

noun, plural minimums, minima [min-uh-muh] (Show IPA)

1. the least quantity or amount possible, assignable, allowable, or the like.

2. the lowest speed permitted on a highway.

3. the lowest amount, value, or degree attained or recorded.

4. an arbitrary amount set by a restaurant, nightclub, etc., as the least amount to be charged each person for food and drink. Compare cover charge.

5. Mathematics. Also called relative minimum, local minimum. the value of a function at a certain point in its domain, which is less than or equal to the values at all other points in the immediate vicinity of the point. Compare absolute minimum. the point in the domain at which a minimum occurs.

The word Minimal comes from the word minimum consisting of synonyms nominal and minimum.

The word Minimum originates from the Latin word minimus meaning smallest or least. Minimum's synonyms consist of minimal and merest.

Based on your sentences above, The correct way of using the words minimum and minimal would be as follows:

The minimal angle-correct because it's being used as an adjective

The minimum angle-incorrect usage using a noun as an adjective

(The same would follow for optimum/optimal)

In mathematics, in particular in combinatorics, these words have a very specific contrastive usage as adjectives.

• A minimal solution to a problem can't be made any smaller by "shrinking" it. If we shrink it, it is no longer a solution. There may be other, distinct smaller solutions though.

• A minimum solution to the problem has the smallest possible size among all solutions. No smaller solutions exist.

This is a very specific, technical usage in certain branches of mathematics. It does not apply to the everyday use of these words.

For example, take these numbers connected with arrows (i.e. a "graph", another technical term):

There are ways to go around in a closed cycle by following the arrows, e.g. 4 -> 7 -> 9 -> 10 -> 5 -> 4:

Which arrows do we need to remove so that there are no such cycles left?

For example, we could remove 6 -> 2, 10 -> 5, 4 -> 7 to break all cycles. This is a minimal solution because not removing any of these three would leave some cycles intact. Thus the solution can't be made smaller. However, it is not a minimum solution because smaller solutions exist. 5 -> 4, 6 -> 2 would be a minimum solution.

In general, all minimum solutions are also minimal, but the converse is not true.

The problem I described above is called the Feedback Arc Set Problem. You will find several such usages of the words minimal and minimum on its Wikipedia page.

In mathematical terms generally minimum means the lowest possible, it is unique. Minimal on the other hand can be sub-optimal.There can be several minimal solutions but only one minimum solution.