Why is ATP hydrolysis to ADP, rather than ADP hydrolysis to AMP, used to drive biochemical reactions?
Summary

The original question was edited — with the approval of the poster — so that in brief it now asks:


  Why is ATP hydrolysis to ADP, rather than ADP hydrolysis to AMP, the
  normal way in which cells drive reactions which alone involve positive changes in Gibbs 
  Free Energy?


The scope of this is wider than that originally assumed in the answer from @user1136, but he has also contributed to this answer by his comments, which I have incorporated into my revision.

To my knowledge there is no generally-accepted provable answer to this question (as opposed to listing advantages for the system as it works today). What I shall do is present a hypothesis that could explain this situation. The fundamental idea† of this is that:


  The contemporary use of ATP rather than ADP in energy transfer
  reactions evolved from a preference for NTPs, rather than NDPs, as the
  precursors of RNA synthesis. This preference was a result of the fact
  that the conversion of NTPs to NMPs and pyrophosphate could prevent
  the reversal of this (and other) synthetic processes in a way that conversions  liberating phosphate could not.


AMP, ADP and ATP: Standard Free Energies of Hydrolysis



There is general agreement that the values for ∆G°′ (standard free energy change at 37˚C) for the hydrolysis of the terminal phosphate bond of ATP (β-γ) and ADP (α-β) are similar. (The following are from J.L. Jain, et al.:

ATP → ADP + Pi  –7.3 kcal/mol
ADP → AMP + Pi  –7.3 kcal/mol


Hence, there is no intrinsic energetic difference that might explain the preferential use of ATP over ADP as an ‘energy donor’ in biochemical processes, i.e. it is not evident why e.g.:


  Glucose + ATP → Glucose 6-phosphate + ADP


and not


  Glucose + ADP → Glucose 6-phosphate + AMP


And, as @user1136 mentions, there are examples where the energy of the hydrolysis of ADP to AMP can be used to ‘drive’ other reactions. He mentions polynucleotide phosphorylase, to which I would add  adenylate kinase, as this is not a synthetic process and is pertinent to the energy metabolism of nucleotide phosphates: 


  ADP + ADP ⇄ AMP + ATP


The introduction of AMP into the discussion raises the question of the value of the ∆G°′ for the following reaction:

ATP → AMP + PPi


Many texts quote a value only slightly greater (ignoring the minus sign) than those for phosphate release (the one cited quotes –7.7 kcal/mol), but a 1995 review of the situation in the Journal of Biochemistry suggests that the difference is actually greater: –10.9 kcal/mol, as opposed to –7.8 kcal/mol for the hydrolysis of ATP to ADP and Pi.

The relevance of this is considered in the next section.

Biosynthetic processes and ATP

It seems to me that the example of polynucleotide phosphorylase is especially pertinent because it is a biosynthetic process in which the nucleotide monophosphate is actually incorporated into the product. 


  [AMP]n + ADP → [AMP]n+1 + Pi


However this ADP-driven non-template dependent synthesis of polynucleotides is not model for template-dependent RNA synthesis, one reason being that it can also operate in the reverse direction as a phosphorolytic 3ʹ to 5ʹ exoribonuclease reaction:


  [AMP]n+1 + Pi → [AMP]n + ADP


Irreversibility is an important feature of template-dependent synthesis of RNA¶, and it is thought that the key to this is that fact that ATP is the substrate and is cleaved between the α and β phosphates liberating pyrophosphate rather than monophosphate.


  [NMP]n + NTP → [NMP]n+1 + PPi


The reason generally proposed to explain why this reaction is irreversible is that in the cell pyrophosphatase hydrolyses the pyrophosphate to phosphate in an exothermic reaction. However the problem in evolutionary terms is that pyrophosphatases would have to evolve before this would work. However, if the ∆G°′ for pyrophosphate release has a greater absolute value than that for phosphate release, this in itself would affect the equilibrium of the reaction in a manner favourable to synthesis, albeit at the greater effective cost of an extra ATP. The later evolution of pyrophosphatases would have served to reinforce and enhance the irreversibility.

The Hypothesis

The hypothesis is that nucleoside tri-phosphates developed early in evolution to provide a means of making nucleic acid synthesis essentially irreversible by a mechanism that generated pyrophosphate. As the example of polynucleotide phosphorylase shows, nucleoside di-phosphates do not allow this.

A priori it would seem that there is no reason why ATP rather than ADP should be preferred for coupling biochemical reactions. However, if one accepts the argument that ATP synthesis was required for nucleic acid synthesis, there would seem no reason for having a set of reactions generating ADP for other purposes. The energy of hydrolysis of ATP to ADP in such reactions would, in any case, have to be utilized, otherwise it would represent a net loss. Why go any further?

It may have been that ATP was preceded in evolution by an an earlier ‘energy donor’. However I would suggest that once a mechanism evolved for synthesizing and using ATP (and other NTPs) for synthetic purposes, it was more efficient to use this for metabolic purposes, largely displacing any previous energy donor molecule.

† Footnote 1

I have touched on this very tangentially in my answer to a quite different question about ATP. The common idea, however, is that the purine and pyrimidine rings of NTPs serve no purpose in energy transfer, but are a relic of their requirement in RNA synthesis, where the NTPs are not merely providing energy, but are incorporated into the RNA product.

¶ Footnote 2

RNA and DNA synthesis are not the only synthetic processes in which this strategy for irreversibility is adopted. The tRNA amino-acylation reaction — essential for protein synthesis is as follows:


  Amino Acid + ATP → Aminoacyl-AMP + PPi


And the first step in glycogen synthesis is the formation of UDP-glucose:


  Glucose 1-phosphate + UTP → UDP-glucose + PPi