The question "What is the relation between lambda calculus and LF?" is comparable to "What is the relation between F (any formal system of representation with an 'alphabet' and a set of 'formation rules' that can recursively generate the expressions of LF) and LF?, or even to "What is the relation between a natural language L and the LF of L?, and the answer is rather trivial: the relation is usually understood to be 'representational' (basically: one of 'translation'), i.e., the lambda calculus can (among many other things!) represent the Logical Form of natural languages, as you know very well.
However, what I suppose you want to understand better is under what assumptions and in what sense can lambda calculus be considered a better representational system for the LF of natural languages than, say, Predicate Logic, or Predicate Logic 'enriched' with events (plus tense logic, modal logic, possible words, etc.). Such formal languages are more or less 'powerful' (and more or less flexible and adequate) to represent the semantic structure of natural languages in a 'compositional' way to the extent that they acknowledge the existence of additional 'entities' (events, time intervals, etc., and new variables that can be quantified over).As you know, the most austere ontologies acknowledge only individuals (and sets) as in first-order Pred Logic, but Russell, Church, Reichembach and others 'enriched' that ontology with 'higher types', then Davidson added 'events', and others even more generously acknowledged 'times' or time 'intervals', modalities, propositions, possible worlds, etc. Montague, of course, has it all. You can take 'lambda calculus' as a sort of limit, in this respect, since, in principle, due to the properties of functional analysis, it can acknowledge and operate with an infinite number of types of entities.
The enormous advantage of lambda calculus over elementary logics as a matelanguage for the compositional representation of natural language LF comes from the fact that, assuming the analyst accepts a very rich ontology (e.g.,Frege's, for example, in which entities are either 'functions' or 'objects' of all types it may be analytically convenient to acknowledge!), it is possible to consider any non-atomic expression 'E' of any NL as the 'product' of two 'factors', a function 'F' and its 'argument' 'a', and say that F yields E if 'applied to' 'a'. What's more, since any function can be trivially turned into an argument of a higher order function ('Type Shifting', 'Type Raising', 'Function Composition'), the analyst can recursively 'reanalyse' functions as arguments (and viceversa!) at will, depending on the type of the 'domains' of the functions he may be interested in acknowledging in order to exhaustively analyze E into just two 'factors', an F and its argument.
And by 'any non-atomic expression E', I literally mean 'any expression', i.e., not just sentences, words, or intermediate phrases of any level of internal complexity (NPs, VPS, APs, AdvPs, PPs, QPs...) but also STRINGS (continuous or not!) of linguistic material that under other analytical approaches (e.g., phrase structure grammar, with its constituency tests) would not be acknowledged as relevant syntactic constituents and therefore would not be eligible to undergo operations (either in syntax or at LF). Actually, since lambda calculus is based on functional analysis and the latter is so powerful a tool, it can in principle be applied even to sub-morphemic units. In other words, 'features' (= [attribute:value] pairs) or even feature-values can also be analysed as 'arguments' (or functions!) if necessary, although, in practice, most syntactic-semantic theories treat minimal 'signs' as unanalysable atoms and seldom or never perform sub-atomic analysis (the event structure of verbs, mainly, excluded).
That makes lambda calculus an extremely powerful and flexible metalanguage for the representation of the 'factors' that compositionally contribute meaning to NL expressions of any level of complexity. You can take the sense of a DP expression like 'the definition of syntactic categories in English' and analyze it into a function 'the definition of syntactic categories in __' and an argument 'English', just as easily as you can take the function to be 'the' and the argument the rest of that DP (the NP), or the function to be 'the definition of syntactic __ in English' and the argument to be 'categories', etc. In principle, given an LF object O you can 'abstract' ANY component C of O and analyse the remainder as a function F( ) = O.
One of the consequences that have attracted more attention is that if lambda calculus is available to compute LF, grammars can be monostratal, i.e., it is possible to compute 'meaning' directly from 'surface syntax'. Thus, constructions that in early PSG or 'Chomskian generative grammar' are supposed to involve 'displacement' (e.g., WH-Movement, Topicalisation, Subject Raising, Extraposition, Right Node Raising, etc.) can be computed as if everything was 'in situ' provided certain kinds of PSG (GPSG, HPSG, etc.) or Categorial Grammar work in tandem with Lambda Calculus to generate syntactic structures and calculate compositional LFs, respectively. Also, since in lambda calculus virtually anything can be treated as a syntactic/semantic 'factor' ('constituent'), coordinative or elliptical constructions (e.g., Gapping, Sluicing, Right Node Raising) that under standard PSG approaches create constituency paradoxes, no longer do in Categorial Grammar cum Lambda Calculus. Mark Steedman's Surface Structure and Interpretation(MIT 1996) or The Syntactic Process (MIT 2000) are authoritative and readable classical statements in this respect. At a more elementary level (and under a Chomskian view of syntax), I. Heim & A. Kratzer's Semantics in Generative Grammar (Blackwell, 1998) is also worth reading.
Of course, all that immense analytical and computational power has its negative side, too: grammars must not only be simple and elegant, but also, and above all, empirically adequate, which means able to generate enough without overgenerating, and, as you can imagine, unrestricted categorial grammar cum lambda calculus overgenerates massively, it acknowledges wildly extravagant LF/semantic entities ('objects' and 'functions') which must ultimately correlate with just as extravagant extralinguistic 'entities', and, needless to say, raises formidable linguistic-theoretic, metaphysical and philosophical issues.
