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 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.

To me, minimum indicates a single, smallest value, where minimal indicates a range of values that are approaching the minimum (and may in fact be the minimum value).

In the same vein, optimum is a single best solution/value, where optimal is a set of solutions/values approaching and including the optimum solution.