There are two parts to this question: why does English use iambic meter while French doesn't, and why does English have 10 syllables in each line of iambic pentameter, while French has 12 syllables per line in an alexandrine. To answer the second question first, it may be pure chance that French settled on 12 syllables per line, while English settled on 10. There are lots of English poems in iambic meter with 8, 10, or 14 syllables per line, and they all work fine. And the dominant meter in French was the decasyllable before it was displaced by the alexandrine in the 17th century.
The reason that French doesn't use iambic meters is because it's almost impossible to write iambic poetry in French.
Before I can explain any further, I need to say something about stress patterns in English and French and some of the rules that govern alexandrines.
In English, every word has some accented syllables, and poetry has to accommodate these. You can emphasize words by using stress, but these emphases fall on top of the phonemic stress patterns already present within the words. In French, there is no stress pattern associated with the words; the stress accents are associated with phrases, and always fall on the last syllable of every phrase. (Except for mute e's, which are never accented. Most of these are pronounced in songs and poems, but only a relatively small fraction are pronounced in usual speech.) So if a word is pronounced alone, its last syllable always has an accent, but if it's in the middle of a phrase, it probably won't.
The alexandrine has 12 or 13 syllables in every line, depending on whether they end with a mute e. The lines are divided into two halves (hemistiches) of 6 syllables each. There is always an accent on the 6th and 12th syllables of each line, and there usually is one other accented syllable in each hemistich. To give an example, from Baudelaire's Chant d'automne (Autumn Song), the first two lines are:
Bientôt nous plongerons // dans les froides ténèbres;
Adieu, vive clarté // de nos étés trop courts!
Shortly we will plummet // into the cold darkness;
Farewell, vivid brilliance // of our summers too short!
Here I've tried to choose words that have the same number of syllables to translate the French, and I've bolded the accented syllables in French and the corresponding syllables in English.
For an entirely English illustration, here is Roy Campbell's translation of the first two lines into iambic pentameter, followed by my translation of the first two lines into an English version of the French alexandrine.
Soon into frozen shades, like leaves, we'll tumble.
Adieu, short summer's blaze, that shone to mock.
We will soon be engulfed // by the cold and the dark.
Dear summer, fare thee well! // Your visit was too short.
Note that this last line would be iambic hexameter if you put stress on the words fare and was. But you wouldn't if it was a French alexandrine. When reading alexandrine poetry, there is a strong tendency to parse it so exactly two syllables are stressed in each hemastich (although this isn't always possible). Similarly, when reading iambic poetry, there is a strong tendency to put some stress on every other syllable, even those (like was in the example above) that wouldn't be stressed ordinarily. And again, this isn't always possible.
So to get a line of iambic verse in French, it needs to consist entirely of two-syllable phrases. This is quite a bit shorter than French phrases usually are. Lines of iambic hexameter are rare in French poetry. The penultimate line of part I of the above poem, Chant d'automne is one. Baudelaire quite deliberately wrote it as iambic hexameter, and I am sure it took some effort his part:
Pour qui? — C'était hier l'été; voici l'automne!
For whom? — It was yesterday summer; look: here's autumn!
(Again, I'm bolding the English corresponding to the accented syllables in the French.)
Edit: I've had a few more thoughts about why alexandrines came to be dominant in French poetry. French poetic meters are generally composed from parts with an even number of syllables, almost always 4, 6, and 8. Decasyllables are broken into 4+6 syllables, while alexandrines are broken into 6+6. Thus, in some sense, alexandrines are more symmetric, and this may be more pleasing. (Although maybe I'm wrong about this, since decasyllables were more dominant for several centuries, although alexandrines were also used during this period.)