Merriam-Webster's Dictionary of Synonyms (1984) puts both words in a group that also includes terse, succinct, summary, pithy, and compendious:

concise, terse, succinct, lconic, summary, pithy, compendious are comparable when meaning briefly stated or presented or given to or manifesting brevity in statement or expression. A person is concise who speaks or writes briefly; a thing is concise that is brief because all superfluities have been removed and all elaboration avoided [examples omitted] ... A person or thing is laconic that is characterized by such succinctness as to seem curt, brusque unperturbed, or mystifying [examples omitted]

S.I. Hayakawa, Choose the Right Word: A Modern Guide to Synonyms (1968), however, places concise in a group with compact, compressed, condensed, constricted, dense, solid, and miniaturized, but places laconic in a group with terse, compendious, pithy, sententious, and succinct. Here is Hayakawa on the "compact" family and, more particularly, concise:

These words refer to confinement or weightiness in a relatively small space. ...

 

Concise, condensed, and compressed suggest progressively greater confinement in space; one use of this progression is to suggest brevity in writing or a lack of wordiness. Concise suggests the use of exactly as many words as are required to express something and no more; it would apply more usually to technical, factual work, but may apply to imaginative prose: [examples omitted].

And here is Hayakawa on the "terse" family and, more particularly, laconic:

These adjectives stress brevity in speech or writing, the avoidance of any wasted words. ... Laconic literally means like a Spartan, with reference to the habitual terseness of Spartan speech. A laconic speaker is so sparing with words as to seem stingy or exceptionally self-controlled. [Examples omitted.] Both a terse and a laconic remark may be so brief as to seem curt. But where a terse remark is complete and its brevity may be due to the pressure of circumstances, a laconic remark may be puzzling and may suggest a deliberate taciturnity: [examples omitted].

In short, both Merriam-Webster and Hayakawa note the fitness of concise to a situation where brevity is appropriate, and view laconic as describing speech that may border on or cross over into rudeness (as curt does). However, Hayakawa also sees laconic as potentially being motivated by a constitutional aversion to lengthy self-expression, whereas concise simply matches the length of the wording to the needs of clarity—and doesn't use a word more.

Merriam-Webster's Dictionary of Synonyms (1984) puts both words in a group that also includes terse, succinct, summary, pithy, and compendious:

concise, terse, succinct, lconic, summary, pithy, compendious are comparable when meaning briefly stated or presented or given to or manifesting brevity in statement or expression. A person is concise who speaks or writes briefly; a thing is concise that is brief because all superfluities have been removed and all elaboration avoided [examples omitted] ... A person or thing is laconic that is characterized by such succinctness as to seem curt, brusque unperturbed, or mystifying [examples omitted]

S.I. Hayakawa, Choose the Right Word: A Modern Guide to Synonyms (1968), however, places concise in a group with compact, compressed, condensed, constricted, dense, solid, and miniaturized, but places laconic in a group with terse, compendious, pithy, sententious, and succinct. Here is Hayakawa on the "compact" family and, more particularly, concise:

These words refer to confinement or weightiness in a relatively small space. ...

Concise, condensed, and compressed suggest progressively greater confinement in space; one use of this progression is to suggest brevity in writing or a lack of wordiness. Concise suggests the use of exactly as many words as are required to express something and no more; it would apply more usually to technical, factual work, but may apply to imaginative prose: [examples omitted].

And here is Hayakawa on the "terse" family and, more particularly, laconic:

These adjectives stress brevity in speech or writing, the avoidance of any wasted words. ... Laconic literally means like a Spartan, with reference to the habitual terseness of Spartan speech. A laconic speaker is so sparing with words as to seem stingy or exceptionally self-controlled. [Examples omitted.] Both a terse and a laconic remark may be so brief as to seem curt. But where a terse remark is complete and its brevity may be due to the pressure of circumstances, a laconic remark may be puzzling and may suggest a deliberate taciturnity: [examples omitted].

In short, both Merriam-Webster and Hayakawa note the fitness of concise to a situation where brevity is appropriate, and view laconic as describing speech that may border on or cross over into rudeness (as curt does). However, Hayakawa also sees laconic as potentially being motivated by a constitutional aversion to lengthy self-expression, whereas concise simply matches the length of the wording to the needs of clarity—and doesn't use a word more.

Merriam-Webster's Dictionary of Synonyms (1984) puts both words in a group that also includes terse, succinct, summary, pithy, and compendious:

concise, terse, succinct, lconic, summary, pithy, compendious are comparable when meaning briefly stated or presented or given to or manifesting brevity in statement or expression. A person is concise who speaks or writes briefly; a thing is concise that is brief because all superfluities have been removed and all elaboration avoided [examples omitted] ... A person or thing is laconic that is characterized by such succinctness as to seem curt, brusque unperturbed, or mystifying [examples omitted]

S.I. Hayakawa, Choose the Right Word: A Modern Guide to Synonyms (1968), however, places concise in a group with compact, compressed, condensed, constricted, dense, solid, and miniaturized, but places laconic in a group with terse, compendious, pithy, sententious, and succinct. Here is Hayakawa on the "compact" family and, more particularly, concise:

These words refer to confinement or weightiness in a relatively small space. ...

Concise, condensed, and compressed suggest progressively greater confinement in space; one use of this progression is to suggest brevity in writing or a lack of wordiness. Concise suggests the use of exactly as many words as are required to express something and no more; it would apply more usually to technical, factual work, but may apply to imaginative prose: [examples omitted].

And here is Hayakawa on the "terse" family and, more particularly, laconic:

These adjectives stress brevity in speech or writing, the avoidance of any wasted words. ... Laconic literally means like a Spartan, with reference to the habitual terseness of Spartan speech. A laconic speaker is so sparing with words as to seem stingy or exceptionally self-controlled. [Examples omitted.] Both a terse and a laconic remark may be so brief as to seem curt. But where a terse remark is complete and its brevity may be due to the pressure of circumstances, a laconic remark may be puzzling and may suggest a deliberate taciturnity: [examples omitted].

In short, both Merriam-Webster and Hayakawa note the fitness of concise to a situation where brevity is appropriate, and view laconic as sometimes bordering on or crossing over into rudeness (as curt does). However, Hayakawa also sees laconic as potentially being motivated by a constitutional aversion to lengthy self-expression, whereas concise simply matches the length of the wording to the needs of clarity—and doesn't use a word more.

Merriam-Webster's Dictionary of Synonyms (1984) puts both words in a group that also includes terse, succinct, summary, pithy, and compendious:

concise, terse, succinct, lconic, summary, pithy, compendious are comparable when meaning briefly stated or presented or given to or manifesting brevity in statement or expression. A person is concise who speaks or writes briefly; a thing is concise that is brief because all superfluities have been removed and all elaboration avoided [examples omitted] ... A person or thing is laconic that is characterized by such succinctness as to seem curt, brusque unperturbed, or mystifying [examples omitted]

S.I. Hayakawa, Choose the Right Word: A Modern Guide to Synonyms (1968), however, places concise in a group with compact, compressed, condensed, constricted, dense, solid, and miniaturized, but places laconic in a group with terse, compendious, pithy, sententious, and succinct. Here is Hayakawa on the "compact" family and, more particularly, concise:

These words refer to confinement or weightiness in a relatively small space. ...

Concise, condensed, and compressed suggest progressively greater confinement in space; one use of this progression is to suggest brevity in writing or a lack of wordiness. Concise suggests the use of exactly as many words as are required to express something and no more; it would apply more usually to technical, factual work, but may apply to imaginative prose: [examples omitted].

And here is Hayakawa on the "terse" family and, more particularly, laconic:

These adjectives stress brevity in speech or writing, the avoidance of any wasted words. ... Laconic literally means like a Spartan, with reference to the habitual terseness of Spartan speech. A laconic speaker is so sparing with words as to seem stingy or exceptionally self-controlled. [Examples omitted.] Both a terse and a laconic remark may be so brief as to seem curt. But where a terse remark is complete and its brevity may be due to the pressure of circumstances, a laconic remark may be puzzling and may suggest a deliberate taciturnity: [examples omitted].

In short, both Merriam-Webster and Hayakawa note the fitness of concise to a situation where brevity is appropriate, and view laconic as describing speech that may border on or cross over into rudeness (as curt does). However, Hayakawa also sees laconic as potentially being motivated by a constitutional aversion to lengthy self-expression, whereas concise simply matches the length of the wording to the needs of clarity—and doesn't use a word more.