Ask Mr. Smarty Plants

You can find a key that lists the differences between two of the three species that you mention— Mimosa strigillosa (Powderpuff) and Mimosa roemeriana (Roemer's mimosa) —plus several other species—in the online version of Shinners & Mahler's Illustrated Flora of North Central Texas on p. 677-682.   Mimosa microphylla (Catclaw sensitive briar) is not included in that volume.

From Shinners and Mahler's key, M. strigillosa is described as:

"Plants with numerous weak, hair-like or bristle-like structures not painful to the touch; recurved prickles absent; mature fruits 10-20 mm long, noticeably flattened, with numerous suppressed hairs on the surface."

M. roemeriana is described as:

"Plants conspicuously armed with numerous recurved prickles capable of causing pain."

This also applies to: 

Mimosa nuttallii (Nuttall's sensitive-briar), Mimosa latidens (Kairn's sensitive-briar) and Mimosa hystricina (Porcupine mimosa)

You can read how Shinners and Mahler's separates these four species.

These four species, plus Mimosa microphylla, were once classified in the genus Schrankia.  A part of the general description (according to Correll and Johnston's Manual of the Vascular Plants of Texas, pp. 775-6) of the genus Schrankia is:

"...armed with numerous recurved prickles (or very rarely the prickles absent)..."

Their synonyms under that genus are:

Synonym for M. roemeriana is:  Schrankia roemeriana.

Synonym for M. nuttallii is:  Schrankia nuttallii.

Synonym for M. latidens is:  Schrankia latidens

Synonyms for M. hystricina are:  Schrankia hystricina and Schrankia nuttalli var. hystricina.

Synonyms for M. microphylla are:  Schrankia angustata, Schrankia microphylla and Schrankia uncinata.

The five species are then separated by:

• "Leaflets with raised reticulate veins beneath" and "Legume 1-4 cm. long, rounded at apex; peduncles 4-12 cm. long; flower heads in early bud with protruding bracts." S. hystricina = M. hystricina.
• "Leaflets with raised reticulate veins beneath" and "Legume 4-12 cm. long, acute or beaked at apex; peduncles 2-7 (-10) cm. long; flower heads in early bud with bracts completely hidden."  S. uncinata = M. microphylla.
• "Leaflets smooth or the midvein prominent only" and "Lower portion of stem distinctly 4- or 5-sided, glabrous; stipules small, 1-3 (-4) mm. long; pod 4-sided, scarcely flattened at maturity; plants of southern Texas."  S. latidens = M. latidens.
• "Leaflets smooth or the midvein prominent only" and "Lower portion of stem rounded, not distinctly 4- or 5-sided; stipules 3-6 mm. long; pod various" and "Mature legume 4-7 (-8) cm. long, flattened, the valves 3 to 6 times as wide as the thickened margin; pinnae 2 to 5 (or 6) pairs; leaflets mostely obtuse at apex; plants of central and north-central Texas."  S. roemeriana = M. roemeriana.
• "Leaflets smooth or the midvein prominent only" and "Lower portion of stem rounded, not distinctly 4- or 5-sided; stipules 3-6 mm. long; pod various" and "Mature legume 6-12 cm. long, tetragonal, not conspicuously flattened or if so the valves less than twice as wide as the margin: pinnae (3 or) 4 to 8 pairs; leaflets mostly acute at apex (rarely obtuse throughout); plants of eaternmost or westernmost Texas" and "Stems, ovary and pod completely glabrous; plants of east Texas."  S. microphylla = M. microphylla.

This is probably a lot more information than you wanted, but you can see that it can be complicated to distinguish between the species.   To complicate things a bit more, the synonyms for M. microphylla have two different descriptions and I wasn't able find the current description for M. microphylla.

Since you live in Travis County I can tell you that M. roemeriana is the species you are most likely to encounter in the county.  Local botanist, Bill Carr, has also reported M. strigillosa and M. nuttallii as occurring here. 

In case you could use some help with the botanical terminology, we have a Glossary of Botanical Terms you can consult online.