This seems an extraordinary claim, given that English is generally considered to have the largest number of words of any modern language (mostly because we just steal them).
Word-use is roughly Zipf-distributed [0] in most all modern languages, so there's not much difference in the number of commonly used words (as long as you ignore basic differences like the existence of articles and the richness of word inflection).
An analysis of English and French texts on Project Gutenburg shows that 93 words account for half of English usage, versus 89 for French [1]. English has fewer words in the middle of the distribution (696 versus 795 at 70%) and in the near tail (6,428 versus 9,050 at 90%, and 14,736 versus 21,231 at 95%), but more in the extreme tail.
So about 50% more words for French. The very extreme tail is less interesting because it captures the size of the dictionary, and words very rarely used.
I think the exponent of the tail would be the most relevant metric, but I can't open those pdf. Can someone plot the inverse CDF and make a log-log plot?
I believe that's the difference between "uses a wider vocabulary" and "has a wider vocabulary".
I would think that literary French generally uses a wider vocabulary than English though, at least because repeating words in French is to be avoided at all costs, while it's more acceptable in English.
After a point, having more words in a dictionary does not necessarily result in the average person using more words in their communication. I would think that education/environment would have a bigger effect.
That said, I would interested if the grandparent could provide more info supporting their statement, as I don't know the truth one way or the other and would like to know more.
This is mostly based on my own observations, I don't have a very specific study on the matter.
Here's a data point though. Many French students learn in English class, as an anecdote, that "shallow" has no equivalent in French. If there were many other words like this, there would be little point in relating this anecdote. "Shallow" is one of the very few words in English with no French equivalent. French words with no English equivalent? I encounter them all the time.
There are different words for the mouth and leg of an animal versus human. (gueule et patte vs bouche et jambe). There's a common word for the lump of flours that form in dough (grumeau). There's a word for a sandwich with a slice of bread on only one side (tartine). There's a single word for brown sugar (cassonade). There's a different word for hair on your head and hair on your body (cheveux, poils), etc.
English words with no French equivalent? Shallow and Serendipity come to mind. I'm googling but I can't find many other at the moment, apart from very technical terms.
English is a bit more subtle. It has inherited a lot of French vocabulary from the Normans, as well as retaining Anglo-Saxon roots. So it often has more words for things other languages have fewer words for, with English having more shades of meaning.
Take happy for example: joyous, joyful, gay, merry, gleeful, etc. I counted some 50 words in the thesaurus under happy, but many of them only translate to joyeaux or gai according to my dictionary checks (although there are also plenty for French, not as many).
English tends to have a lot of words that are close synonyms but with different evocations. It's possible to converse English in language almost completely of French origin, or speak plainly in the common tongue with older words. The former sounds flowery; the latter tends to remind one of the Bible.
Many English dictionaries have more than 300k entries. The reason is that many of the words are close synonyms inherited from different sources. That's what I was trying to get at in my carefully constructed sentence.
IMO the synonyms you link to aren't as close as those at http://www.thesaurus.com/browse/happy - I believe there are further English translations of many of those on your page, but they take things in a different direction. Many of them relate to luck, for example.
I experimented with synonymo.fr a little further. It looks like many distinct English words have a single French word translation, and that French word has different meanings in different contexts. Consequently, looking up the synonyms for that word, you end up seeing synonyms for all the different contextual meanings for that word.
For example, take the word 'sky'. A quick scan seems to suggest French doesn't have distinctly separate words for sky vs heaven, translating both as ciel. Synonymo.fr suggests lots of synonyms, but they are mostly synonyms for heaven, not sky. It's like this with most French words I experimented with; same word, lots of different meanings, where English has separate words without the same degree of "meaning spread". That is, the English words are more precise and less ambiguous. And there are usually more of them, without wandering into different concepts.
This seems an extraordinary claim, given that English is generally considered to have the largest number of words of any modern language (mostly because we just steal them).