The argument that the English "will + infinitive" construction should not be considered a future tense is fairly complex. It is not an obvious matter, and I think the rejection of this classification is usually based on several criteria, not just one.
I believe a tense is usually defined as something like a grammaticalized construction used for time reference. I know that there is a lot of linguistic literature about the exact meaning of "tense" and how it works, but I haven't read any of it, so I can't give an explanation of whether the semantics of the English "will" construction qualify as future tense or not from a semantic point of view.
Contrary to what has been said in other posts and comments, I don't think there is any general consensus that tense can only be expressed by synthetic, and not by analytic constructions. French, German and Latin all have analytic constructions for some tense/aspect/voice combinations, but these constructions are still generally understood to be marked for tense (e.g. French "il est allé" is typically classified as a past tense construction despite the fact that the finite verb, the auxiliary "est", does not bear past-tense inflection).
I'm also not sure whether it is supportable to say that some other languages have analytically expressed tenses, but English specifically does not. If I remember correctly, the CGEL analysis of the English tense system categorizes the perfect as a secondary axis of tense (rather than as a primarily aspectual construction), even though the perfect is formed analytically.
Optionality and frequency (I think as proxies for grammaticalization) are criteria that can be applied even to languages with a synthetic "future" construction
I think Draconis's answer, in bringing up the topic of "optional" vs "mandatory" constructions, gets at a more important point. Tense is supposed to be "grammaticalized", and in general, more grammaticalized categories in a language are expected to be more obligatory and to have fewer competing alternatives for their expression than less grammaticalized categories. I'm not sure I put this very clearly, so here is an example of what I mean: English the is considered to be a highly grammaticalized word, of a type that has been given a special name ("definite article") and that is often viewed as distinct from the less frequent demonstratives this and that. In some languages, demonstratives can be used in certain contexts to express a similar idea to that of the English definite article. But if it is not obligatory to mark definite noun phrases with a demonstrative, the language is less likely to be categorized as "having definite articles".
In fact, even languages with synthetic constructions that refer to the future are not invariably classified as having future tenses. In the comments section of the World Atlas of Language Structures chapter 67, "The Future Tense", by Östen Dahl and Viveka Velupillai, Dahl mentions an argument for not classifying Portuguese's inflectional future construction as a future tense because the verb form is "not [...] used generally enough to be treated as a grammaticalized future tense (it is in competition both with the present and with the periphrastic ‘go’ construction)."
Dahl and Velupillai also restrict their chapter to synthetic/inflectional constructions, but I feel like this criterion may have been chosen in part for convenience.
I found a paper by Martin Haspelmath about the issue of classifying certain analytic constructions as "periphrastic" ("Periphrasis", Article 68 of Morphology: A Handbook of Inflection and Word-Formation, HSK, de Gruyter) which may have some relevance. Haspelmath brings up the connection of the concept of "periphrasis" and grammaticalization, and also draws a distinction between "suppletive periphrasis" of the kind found in the Latin tense paradigm (where some kinds of futures are expressed analytically, but others are expressed synthetically) and "categorical periphrasis" like the English have-perfect, which is always expressed analytically.
To understand the contrast between what English has and what a "real" future tense would be, it might be best to look at some of the following languages
As some comments have noted, many other European languages can be viewed as having the same kind of problematic aspects to their "future tenses": not only Germanic languages, but also Slavic languages and even, as I mentioned above, some Romance languages, despite the development of a synthetic future construction in Romance.
For the sake of better understanding the range of future constructions in languages, and how tenselike they can be, it may be helpful to study the following languages mentioned in another paper by Dahl:
As noted in Bybee and Dahl (1989), the future grams in an expanded version of the sample used in Dahl's earlier investigation (Dahl 1985) which were systematically used in both temporal and conditional clauses were all bound. In the expanded sample, the languages in question were the following: Alawa, Bandjalang (Australian), Oneida, Seneca (Algonquian), Hebrew (Semitic), Hindi/Urdu, Kurdish, Latvian (Indo-European), Georgian (Kartvelian). The futures in these languages are also characterized by a number of other indicators of high degree of grammaticalization: close adherence to the generalized cross-linguistic profile of the gramtype, high frequency of use and tendency to obligatory use in central cases (to the extent that
all these things can be judged about from the questionnaire data).
[...] the absence of any languages from Western or Southern Europe, two areas that are rather over-represented in the sample, should be noted. The conclusion is that full grammaticalization of futures is not common in
large parts of Europe. This is a point that we shall return to.
("The grammar of future time reference in European languages", by Östen Dahl, p. 5)