Since the question posed is about the "In what way does the type of operator or the type of function matter?", I thought the following observation will be apt:

As pointed out by Liviu Nicolaescu in the comment above, Taylor's approach seems to have much wider applicability when it comes to functional calculus. In fact in page 295 of Taylor's book it is mentioned that Seeley's results form a special case of the result.

Moreover, these methods have gone beyond elliptic operators. For instance, Uhlmann, Melrose and Guillemin have developed a framework of distributions whose wavefront sets are in several Lagrangian intersecting manifolds (pseudodifferential operators with singular symbols) for a functional calculus on real principal-type operators, operators of double characteristics, wave operators. Principal symbols also have been computed for these operators and all the computations are purely symbolic.

Since the question posed is about the "In what way does the type of operator or the type of function matter?", I thought the following observation will be apt:

As pointed out by Liviu Nicolaescu in the comment above, Taylor's approach seems to have much wider applicability when it comes to functional calculus. In fact in page 295 of Taylor's book it is mentioned that Seeley's results form a special case of the result.

Moreover, these methods have gone beyond elliptic operators. For instance, Uhlmann, Melrose and Guillemin have developed a framework of distributions whose wavefront sets are in several Lagrangian intersecting manifolds (pseudodifferential operators with singular symbols) for a functional calculus on real principal-type operators, operators of double characteristics, wave operators. Principal symbols also have been computed for these operators and all the computations are purely symbolic.

Since the question posed is about the "In what way does the type of operator or the type of function matter?", I thought the following observation will be apt:

As pointed out by Liviu Nicolaescu in the comment above, Seeley's or Shubin's approach seems to have much wider applicability when it comes to functional calculus. The methods have gone beyond elliptic operators. For instance, Uhlmann, Melrose and Guillemin have developed a framework of distributions whose wavefront sets are in several Lagrangian intersecting manifolds (pseudodifferential operators with singular symbols) for a functional calculus on real principal-type operators, operators of double characteristics, wave operators. Principal symbols also have been computed for these operators and all the computations are purely symbolic.

Since the question posed is about the "In what way does the type of operator or the type of function matter?", I thought the following observation will be apt:

As pointed out by Liviu Nicolaescu in the comment above, Taylor's approach seems to have much wider applicability when it comes to functional calculus. In fact in page 295 of Taylor's book it is mentioned that Seeley's results form a special case of the result.

Moreover, these methods have gone beyond elliptic operators. For instance, Uhlmann, Melrose and Guillemin have developed a framework of distributions whose wavefront sets are in several Lagrangian intersecting manifolds (pseudodifferential operators with singular symbols) for a functional calculus on real principal-type operators, operators of double characteristics, wave operators. Principal symbols also have been computed for these operators and all the computations are purely symbolic.