Standard accounts of Turing Machines in the literature assume an infinitely long tape in at least one direction (and indeed infinitely time long to perform its computations). Clearly in practice no computer could have an infinite amount of cell storage either from the commencement of computations (even if we disregard the infinite amount of available time) or by adding blank cells at the end of the tape as required (which we'll eventually need if we assume infinite amount of time).
So my question is this: do the properties about Turing Machines obtained by assuming an infinitely long tape still hold of practical computing devices, for instance the existence of uncomputable functions? If so what does the assumption of an infinite tape buy us in terms making it easier to prove such properties?
Yes it makes a huge difference! A Turing machine with a finite tape is just a finite automaton, which we know are much less powerful. For example, a TM with a finite tape cannot compute a context free grammar. All of this is in theory, of course.
Let me mention linear bounded automata (LBA) which can compute a proper subset of the function Turing machines can handle. LBA do model real computers better than Turing machines in the sense that no computation can use an infinite amount of space but there is no (constant) bound on space either. Of course, real computer do not have to have a linear bound.