This is a somehow extended comment and complement rather than an answer (I've already upvoted Ashen's answer).
The number of total words in a dictionary or the number of total words for college or basic conversation is sometimes biased by the meaning of word. Let's take one example out of many: “typewriter” v/s «máquina de escribir». The English counterpart is a composed word, the union of the noun “type”, the verb “to write” and the suffix “-er” (something that writes using types), but becomes one word with its own dictionary entry. The Spanish counterpart is also composed by three lexemes (content unit in language, including words, roots and adfixes), just that each lexeme is its own word and there is no individual entry in the dictionary for the concept. A literal translation into English would be “machine for writing”. Same concept, same number of lexemes, yet one dictionary entry more in one language than in the other.
On the other hand, dictionary words are mostly roots: the non-declined form of a word. So in a dictionary you will find the word “dog” but not the word “dogs”, and the word “(to) walk” but not the words “walks”, “walked”, “walking”.
The number of word variants in Spanish is much higher than the number of word variants in English. A regular verb in Spanish has over 50 conjugation forms compared to four different forms in English. Number and gender declinations in adjectives and articles. Gender declination in nouns sometimes adding extra meanings. Pronominal verbs having a shifted meaning without adding a different word as dictionary entry, etc.
Aditionally English has more irregular plurals, that deserve a different dictionary entry than the singular counterpart, and even Spanish irregular verbs have many patterns that dictionary authors do not care to provide most irregular forms as dictionary entries.
So one difference in counting words is what do you define as a word (and how a language works its mapping between lexemes and words) and what you count as different words.
The other difference in dictionary length are dictionary authors. The Spanish language has a very centralized and usually conservative body that regulates what goes into the official dictionary of the Spanish Language: Diccionario de la lengua española by Real Academia de la Lengua (RAE). Independent dictionary authors and brands do not go too far from RAE's dictionary. While RAE was the central Language Academy, until very recent years RAE was mostly a Spanish institution (Spanish as from Spain) while the Language Academies in other Spanish speaking countries were subordinated to the Royal Academy. Very regional words from a certain region in Spani Spain would be more prone to get into the official dictionary (v.g. «altor»), than words with more extensive use in Latin America. Before the ubiquitousness of the Internet era, the process for RAE to admit a new term, either coined in Spanish, borrowed from other language, or slang in any region of the Spanish speaking world, was a slow process, milestoned by editions of the printed work, and centralized in Spain. The process has changed and currently many new words such as «tuit» (tweet) have been accepted.
Dictionary authors in English have usually been more liberal in accepting new words, as they have a more descriptive than prescriptive approach. They want the dictionary users to know what that obscure word found in that obscure text by a English speaking writer means, rather than to tell dictionary users which is the correct form and use of the word.
So, how you count words («máquina de escribir» v/s “type-writ-er”) and how you count distinct words («caminar, camino, caminasa, caminando, caminase, caminó, caminara, caminará, caminado, etc.» v/s “walk, walks, walked, walking”) will make a difference in the number of word needed for college level or casual conversation. An dictionary authorship and criteria will also add a difference in words counted as dictionary entries. All of this without affecting general expressiveness in either language.