Discontinuous Tube visualization

Update: You can use your function TR with RegionFunction:

a = 2; h = 0.4 a; thmax = 10;
ParametricPlot3D[{a Cos[th], a Sin[th],  h th/(2 Pi)}, {th, 0, thmax},
  PlotStyle -> {Red, Tube[.2]}, Axes -> False, Boxed -> False, PlotRange -> All, 
  RegionFunction-> (TR[#4] == .2&)]

Alternatively, use it with ConditionalExpression (or with Piecewise) to change the first argument of ParametricPlot3D:

ParametricPlot3D[ConditionalExpression[{a Cos[th], a Sin[th], h th/(2 Pi)}, TR[th]==.2], 
 {th, 0, thmax}, 
 PlotStyle -> {Red,Tube[.2]}, Axes -> False, Boxed -> False, PlotRange -> All]

same picture

Original answer:

Using MeshFunctions + Mesh + MeshShading:

n = 1000;
ParametricPlot3D[{Cos[th], Sin[th], th/5}, {th, 0, thmax}, 
  PlotStyle -> Yellow, Axes -> False, Boxed -> False, PlotRange -> All,
  BaseStyle -> Directive[CapForm["Butt"], JoinForm["Round"]], 
  MeshFunctions -> {#4 &}, 
  Mesh -> {Subdivide[0, thmax, n]}, 
  MeshStyle -> Opacity[0], 
  MeshShading -> {Red, None}] /. Line -> (Tube[#, .18] &)

Neat Examples:

Play with combinations of CapForm["Butt"]/CapForm[None] and MeshShading -> {Red, None} / MeshShading -> Dynamic @ {RandomColor[], None} to get nice effects like:

With n = 500, change the replacement rule to

 Line -> (Dynamic[Tube[#, RandomReal[{.05, .3}]]] &)

to make the tube radii random to get effects like:

Note: MichaelE2 brought to my attention that in version 12 the code above produces tube segments with rounded caps like this.

Changing MeshShading to

 MeshShading -> {Opacity[.99999, Red], None}

or changing the post-processing rule to

 {Line -> (Tube[#, .18]& ), r_RGBColor :> Opacity[.999, r] }

or to

 Line-> (Tube[#, .18] & /@ Partition[#, 2, 1]& )

fixes this issue (don't know why/how though). Perhaps, the issue reported in this q/a that is supposed to be fixed in v10.2 lingers.

Yet another fix is to add the option

Method -> {"TubePoints" -> 50}

to ParametricPlot3D.

It works for me if you add more PlotPointsand let MaxRecursionwild

ParametricPlot3D[
 {Cos[th], Sin[th], th/5}
 , {th, 0, thmax}
 , PlotStyle -> {Red, Tube[TR[th]]}
 , Axes -> False
 , Boxed -> False
 , PlotRange -> All
 , PlotPoints -> 5000
 , MaxRecursion -> Infinity
 ]