Auxiliary means to help or support something else, to act as a backup, and to be secondary to something more significant. Auxiliary verbs are often called helping verbs.
Lexical means "relating to words." Lexical verbs are the main verbs in a phrase or sentence. To put it simply, lexical verbs are any verbs which are not auxiliary verbs. Lexical verbs carry their own meaning, (full verbs), and therefore can use but don't necessarily need a helping (auxiliary) verb.
- I was acting the jerk yesterday at the carnival.
"Was" is the auxiliary verb and "acting" is the lexical verb because it describes the state or action in the sentence.
- If we go to that restaurant, I will want the lobster.
"Will" is the auxiliary and "want" is the lexical verb.
Some examples of auxiliary verbs are: be, is, was, may, have, had, can, could, would, did, might.
Some examples of lexical verbs, which are much more numerous, are: run, think, see, walk, go, pull, make.
Everyone learns, from an early age, that a verb is an action or "doing" word. As learners are given more information about verbs, they come to recognize distinctions in tense and the nature of verbs; whether they are passive or active, transitive or intransitive, finite or non-finite, simple, continuous or perfect tenses, and even what mood they create; prompting questions about the intention, attitude or "mood" of the verb being expressed.
Auxiliary verbs, affectionately called "helping" verbs often help define tenses, mood and voice but are also often overlooked as they defy the basic tenet of a verb as they do not actually indicate what action may be taking place. Lexical verbs are those easily recognizable action words; the word "lexical" indicating the most common use or intention of, in this case, a verb which, as pointed out, is to denote an action. To clarify, most verbs are lexical verbs and, if not lexical verbs, they are auxiliary verbs.
The biggest difference between the two types of verbs is that lexical verbs indicate the main action taking place in any sentence and therefore the intention of the sentence becomes clear whereas, auxiliary verbs have a more subtle function because they often complete a sentence without the reader being aware how they contribute to the structure of the sentence and, without which, the sentence would make little or no sense. This obviously means that they have very different uses. Lexical verbs are more about the meaning whereas auxiliary verbs are more about their grammatical effect.
The above answers do a good job, but it is good to start with definitions.
Lexical verbs express action, state of being, or predicate meaning. In a word, they are the main verbs of a sentence.
An auxiliary verb is a helping verb, that is, auxiliary verbs help the main verb. When auxiliary verbs exists, there is a verb phrase.
Here are a few examples:
1. The boy ran into the forest. "Ran" is the main or lexical verb.
2. The boy will have run into the forest. "Ran" is still the main or lexical verb, but the words "will have" are auxiliary verbs, as they help the main verb.
3. She saw the bird. "Saw" is the main or lexical verb.
4. She could have seen the bird. The words "could have" are auxiliary verbs.
The term "auxiliary" comes from the Latin word "auxilium", which means "help", especially in a military context. The comparison to the battlefield is useful because it helps us understand that if the auxiliary verbs can be a great asset to the lexical verbs, the battle itself is led by the lexical verbs: they are the lead actors, the auxiliary verbs are mostly backstage, and used if needed.
Auxiliary verbs are commonly used to create the compound verbal tenses and voices and are associated with present or past participles. To that extent, "to be" and "to have" are the only auxiliary verbs per se.
Examples: I am writing, she has done, you have been said...
Some verbs are traditionally called "semi-auxiliary" verbs because they emphasize the aspects and/or the modes. In that case, they are followed by the the infinitive without "to".
The aspectual auxiliary verbs say if an action is done, is currently going or is up to be done. They also say if the action starts or has just been done.
The modal auxiliary verbs mostly express possibility and obligation: I can do it; you must not forget it.
The term "lexical" comes from the old Greek root meaning "word": the lexical verbs are meaningful in themselves, they are "the action" itself, contrary to the auxiliary verbs which put the lexical verbs in context.
As such, the lexical verbs are often the core of a sentence: "you ask, I answer" and are also the large majority of the verbs in the dictionary.
Lexical verbs are also known as main verbs that are words showing state or action. Like;
Auxiliary verbs are also known as helping verbs. Basically these are words that help enhancing the meaning of the verbs. These show tense (past or present), aspects, modality(state) and voice(active or passive). Like;