How are the poets Elizabeth Barrett Browning and Dylan Thomas similar and different in their writing style in the poems "How Do I Love Thee? Let Me Count The Ways" and "Do Not Go Gentle into That Good Night"?

The first area you should compare is prosody. Both poets are using traditional meter. In terms of rhythm, both poems are written in iambic pentameter. Thomas's poem, however, has a greater number of spondaic substitutions than Browning's. One distinctive feature of Thomas's poem that gives it a sort of weight or solemnity is the use of initial spondaic substitutions.
In terms of rhyme structure, Browning's poem is a sonnet and Thomas's a villanelle. Browning uses the Petrarchan or Italian sonnet form of an octave and a sestet, rhymed ABBAABBA CDCDCD. The villanelle form used by Thomas, unlike the sonnet, is divided into stanzas. A villanelle consists of several three-line stanzas rhymed ABA with a final four-line stanza rhymed ABAA. Another major difference between the sonnet and the villanelle is that a villanelle uses refrains, repeating entire lines. The final line of the first stanza is repeated in all subsequent odd-numbered stanzas, while the final line of the second stanza is repeated as the final line of all even-numbered stanzas. The final four-line stanza has two regular lines followed by the first and then second refrain. Browning also uses repetition quite heavily, often repeating the phrase "I love thee," but the repetitions do not form a regular pattern.
While both poems address love and build to a conclusion about the relationship between love and death, Browning's progresses logically and steadily to its conclusion, while Thomas uses the rhetorical technique of crescendo to build to a dramatic conclusion.
In terms of language, both poems rely heavily on monosyllables. Browning is slightly more monosyllabic and uses slightly more abstract language, while Thomas is more vivid, sensual, and concrete. Thomas also uses comparison more heavily than Browning, giving several examples of how other types of men struggle against death (wise, brave, wild, and good men) before addressing his father, while Browning's comparisons illustrate the quality of her love rather than comparing it with that of other people. Browning's poem uses the first person throughout while Thomas only introduces the first person in the final stanza. 
Thomas relies more heavily on assonance and alliteration than Browning. A dramatic example of his use of assonance can be found in the first stanza:

Old age should burn and rave at close of day;Rage, rage against the dying of the light.