Answer: The short answer is that, yes, English has far more words at its disposal than does Spanish, but the ratio isn't anywhere near that large.
The ratio is probably closer to 2:1, although there is no exact way to give an answer. Except perhaps in the case of some minor languages with a very limited vocabulary or obsolete or artificial languages, there is no agreement among authorities about which words are a legitimate part of a language or how to count them. Furthermore, any living language is in a continual state of change. Both Spanish and English are continuing to add words English primarily through the addition of technology-related words and words related to popular culture, while Spanish expands primarily through the adoption of English words.
It is generally stated (although authorities differ) that English has around 250,000 to 300,000 words when obsolete words (which still exist in some unabridged dictionaries) aren't counted. There is one count that puts the English vocabulary at about 1 million words but that count presumably includes words such as Latin species names, prefixed and suffixed words, jargon, foreign words of extremely limited English use, technical acronyms and the like, making the count as much of a gimmick as anything else.
All that said, it is probably fair to say that English has about twice as many words as does Spanish. Large college-level English dictionaries typically include around 200,000 words. Comparable Spanish dictionaries, on the other hand, typically have around 100,000 words. Of course, many of those words are seldom used. Estimates I have seen indicate that Americans typically have a speaking vocabulary of around 20,000 words, although in a typical day they actually use far fewer. I've been unable to find reliable estimates on the everyday vocabulary of Spanish speakers, but I'd guess that the number of words used by educated speakers in a typical day would be about the same in either language. In either language, it is possible to communicate reasonably well with fewer than 1,000 words.
One reason that English has a larger vocabulary is that it is a language with Germanic origins but a tremendous Latin influence, an influence so great that sometimes English seems more like French than it does like Danish, another Germanic language. The merging of two streams of language into English is one reason why we have both the words "late" and "tardy," words often interchangeable, while Spanish (at least as an adverb) has only tarde. The most similar influence that happened to Spanish was an infusion of Arabic vocabulary, but the influence of Arabic on Spanish isn't close to the influence of Latin on English.
The fewer number of words in Spanish, however, doesn't mean that it can't be just as expressive as English; sometimes it is more so. One feature that Spanish has when compared to English is a flexible word order. Thus the distinction that is made in English between "dark night" and "gloomy night" might be made in Spanish by saying noche oscura and oscura noche, respectively. Spanish also has two verbs that are the rough equivalent of the English "to be," and the choice of verb can change the meaning (as perceived by English speakers) of other words in the sentence. Thus estoy enferma ("I am sick") is not the same as soy enferma ("I am sickly"). Spanish also has verb forms, including a much-used subjunctive mood, that can provide nuances of meaning sometimes absent in English. Finally, Spanish speakers frequently use suffixes to provide shades of meaning.
All living languages seem to have the ability to express what needs expressing; where a word doesn't exist, speakers find a way to come up with one whether by coining one, adapting an older word to a new use or importing one from another language. That's no less true of Spanish than of English, so Spanish's smaller vocabulary shouldn't be seen as a sign of inferiority.