The question arose in the comments here.
Linguistic distance is not a really well-defined term. When you consider a tree model of language evolution (this assumption is contested, e.g., by wave models of mutual influence) you can define linguistic distance as the time since the split from the last common predecessor. You can imagine different methods to measure or aestimate this time.
Lexical distance is one of these methods. It assumes a more or less constant replacement rate for core lexical items over all languages (Again, a contestable hypothesis).
An alternative method to Lexical distance measures the amount of sound changes in cognate words (see my answer here).
A problem with a tree model based linguistic distance is that it cannot measure linguistic convergence (e.g., the formation of a Sprachbund).
Lexical distance (based on methods as Levenshtein distance) serves to determine the linguistic distance. It is the first step to elaborate a classification of linguistic varieties (they can be displayed as tree). So lexical distance is tied to linguistic distance. Without that we can use other linguistic facts (phonemes, syntactic order, ...) in order to elaborate other kinds of distance (for example phonemic distance, based on their distinctive features). But the morpheme is the linguistic fact the most used, that is why lexical distance is often associated with linguistic distance. Now, the discipline that studies the linguistic distance is called dialectometry.
Probably (especially it is true for the intelligibility linguistic studies), the lexical distance, as a part of lingustic distance is percent of the words of completely different roots or of not identical meaning between two languages, at least in some written or spoken texts.
The other parts of linguistic distance, as it is correctly and often mentioned here by other participants, are:
a) The orthographical (for a written text) or phonetic distance for a sooken text) of cognates and (in general partial cognates (same-root or close root words) with the same meaning. It is usually mradured by the Normalized Levenshtein distance. These kinds of the linguistic disrance are calculated and shown most often. There are the other kinds of the linguistic distance which are calculated less often in the respective researches:
b)Grammar distances - differences in parts and forms of speech (like additiinal tenses etc.)
c) Syntax distances - ckncerning wird order.
d) morphological distance (something connected to phonetic and orthigraphic distances). It is how big differences in preffixes and suffixes between two cognate and partial cognate words are. Or how, in this case, is big phonetic or orthographic similarity in roots.