Q:

Why is an initiator tRNA required, distinct from the methionine tRNA used in elongation?

I'm confused by why there is a need for different tRNA-methionine complexes for translational initiation and elongation. 
This paper mentions that

It is important that each type of methionyl tRNA be restricted to its separate function, as competition for tRNA by the initiation and elongation machinery could lead to serious problems for the cell

I don't understand what these problems may be. 

A:

I think the key to understanding this is to appreciate how different the initiation process is from the rest of translation. 
The 30S ribosomal subunit recognises start codons via an interaction with the Shine-Dalgarno sequence that lies upstream in the mRNA. The remaining steps of initiation involve recruiting fMet-tRNA and, critically, placing it into the P site of the ribosome. In contrast, during elongation, aminoacyl tRNAs enter at the A site before their amino acid cargo is joined to the growing polypeptide which is held in the P site as a peptidyl tRNA. So you can see that the initiation process is special because of the need to prime the whole translation process by placing the first 'peptidyl' tRNA (actually N-formylmethionyl tRNA) into the P site. 
There are two tRNAs for methionine, tRNAfMet and tRNAMet, and only the former is a substrate for the enzyme methionyl-tRNA formyltransferase. This effectively partitions the supply of methionine into two separate pools, one for use in initiation and one for use in elongation. In fact you could argue that initiation involves a different amino acid (N-formylmethionine) which is also coded by AUG.
Since an N-formylmethionine is essential for initiation, the simplest alternative would be to have a situation where a single species of methionyl-tRNA was a substrate for the transformylase. This would mean that N-formylmethionine would have to be allowed into the A site, and all those formyl groups would have to be subsequently removed from the internal methionines. 
Update in response to comment from the OP
Having re-read the original question, I can see that I jumped the gun and actually answered a different question. The real question is much deeper. 
Firstly, although the details differ, both bacteria and eukaryotes feed methionine through separate pathways for initiation and elongation.  Bacteria are discussed above; eukaryotes channel methionine through two different pools of tRNA, tRNAiMet and tRNAMet. Clearly this channelling is a universal feature of translation systems.
Next, let's look at tRNAMet abundances in E. coli. Since all protein-coding genes have a single start codon and, on average, a few internal methionine codons, we might anticipate that there would be more tRNAMet than tRNAfMet. But this isn't the case: according to Dong et al 1996 tRNAfMet is present at around 3x the level of the elongation species. 
This suggests to me that there is something inherent to initiation that requires a higher level of the corresponding methionyl tRNA. Now you might say that this discrepancy is rather small but bear in mind that one of the properties of Met-tRNAfMet is that is easily rejected by elongation factors (reference). This is important because any charged tRNA can enter the A site before being compared to the current codon. By being easily recognised as 'wrong' for elongation, Met-tRNAfMet is, to some extent, excluded from the elongation process.
So far I have presented facts -  what follows are my own ideas.
If a single Met-tRNA species was used both for elongation and for initiation it would have to optimised for entry into the elongation process, spending time interacting with ribosomal  A sites. In order to maintain the level of free Met-tRNA required to drive initiation I conclude that much higher levels of this single Met-tRNA would therefore be needed. This in turn would increase the transient entry of this species into all A sites, and it would essentially act as an inhibitor of elongation.

A:

The same paper states the following just before the part you have quoted:

The cell acquires an additional degree of control by having a separate tRNA for initiation, and thus regulates the levels of
  initiator and elongator methionyl tRNAs separately.

It is just another factor that controls translation. If there is no initiator tRNA, the ribosome cannot start synthesizing a certain protein. Imagine the cell needs to synthesize e.g. 100 copies of a protein. If initiation and elongation tRNA were the same, it would be much harder to determine how many methionyl tRNAs would be required to synthesize 100 copies, especially since the cell cannot tell how many methionins are required for a certain protein. Instead, if initiator tRNA is separate, 100 initator tRNAs will lead to 100 synthesized proteins (provided the other initiation factors are present, obviously).

A:

Interaction with tRNA-binding factors is why separate tRNAMet species are employed

A specific tRNAMet is required for initiation as the initiating
  met-tRNA must interact with a specific tRNA-binding initiation factor
  (IF2) in order for it to bind to the P-site on the small ribosomal subunit.
  The elongator tRNAs share common structural features that enable them to
  interact with an elongation factor (EF1), which is needed for binding
  to the A-site of the whole ribosome. If the elongator met-tRNAMet were
  able to interact with the initiation factor, so would all the other
  aminoacyl-tRNAs. It is this competition for the initiation factor (and
  the small ribosomal subunit) that a separate initiator tRNAMet avoids.

This answer applies to all modes of prokaryotic and eukaryotic initiation, regardless of the manner in which the ‘correct’ initiation codon is recognized, as the interactions mentioned (shown in the cartoon below for those unfamiliar with this) are universal and ancient.

[E-TBF and I-TBF are my own non-standard abbreviations for Elongation tRNA-binding factor and Initiation tRNA-binding factor, respectively. The several arrows following the formation of the ternary complex are to indicate that several steps (which for initiation may differ between kingdoms — and between mRNAs in a single genome) occur before the final specific interaction of the tRNA anticodon with the appropriate mRNA codon on the appropriate ribosomal species.]
The cartoon representation of the factors and complex are taken from a diagram by Awchen and illustrates the interaction of the bacterial elongation factor (the original name, EF-Tu is used) and aminoacyl-tRNA. The structure of the initiation factor and its interaction with the tRNA is different (as is the conformation of the tRNA). Although not important for this answer, they are shown below in a comparison diagram from Schmitt et al., which unfortunately does not show the same orientation for the tRNA.

Why separate tRNAs, rather than a single multifunctional tRNAMet?
The answer given above describes how — in contemporary biological systems — the existence of two tRNAs for methionine prevents competition from elongator tRNAs for the initiation factor, IF-2, and hence competition for the small ribosomal subunit in its ‘initiation state’. However it is reasonable to ask why a separate initiator tRNAMet was necessary for such discrimination — surely IF-2 could have been specific for a single tRNAMet by recognising the same features that a single met-tRNA synthetase does. (The argument that then there would have been competition for EF-1 seems weak.) This is a distinct question, about which one can only speculate. My observations regarding it follow.
This argument presupposes that in the development of a single AUG initiation codon from what one assumes was a non-specific initiation system, all that was required in the tRNA was an appropriate anticodon — the sequence context and accessory proteins (initiation factors) being responsible for selecting the initiating AUG and triggering the initiation process. However this is evidently not the case in contemporary biological systems. 
The initiator tRNAMet is structurally distinct from the elongator tRNAs. There is a striking reflection of this in the fact that although eukaryotic initiator tRNAMet is not formylated and no transformylase exists in eukaryotic cells, this species — but not the elongator species — can be formylated in vitro by the E. coli transformylase (see, for example, review by Kolitz and Lorsch). Clearly, specific structural features of initiator tRNAMet have been conserved over a long period of time, despite the evolution of different methods of selecting the ‘correct’ initiation codon (Shine and Dalgarno, Kozak scanning, ribosome landing pad).
Because of these latter differences in contemporary initiation it is impossible to know what the ‘original’ system was and why there was a need for structural differences in the initiator tRNA. However one answer is suggested by the fact that in contemporary eukaryotic initiation by the predominant Kozak scanning method the initiator tRNA binds to the small ribosomal subunit in the absence of the AUG codon. One suspects recognition of the P-site on a single subunit was as important as  codon–anticodon recognition.
Given the requirement for major structural changes in tRNAMet for initiation, the development of multi-functionality in single species would seem considerably more difficult than the creation of new species by gene duplication and subsequent mutation. Creation of new tRNAs by gene duplication is a relatively common process, and in this case would have left the original elongator tRNAMet untouched while the duplicate (still containing the recognition features for the met-tRNA synthetase) was free to evolve features which changed its ribosome- and factor-recognition properties.

