In English language, to + verb is used in an infinitival construction (a clause in which the verb is infinite). An infinite verb is a type of non-finite verb (there are two more types of non-finite verbs, namely Gerunds and Participles), which does not bear tense and phi-feature agreement morphology. In other words, unlike a finite verb, the verb in its infinite form does not agree with the subject of the sentence for phi features.
Let us take the example of the verb swim. In its finite form, the verb agrees with the number and person (phi) features of the subject in English. The –s morpheme that appears on the verb is a marker of number and person agreement with the subject. In other words, it appears on the verb only when the subject is 3rd Person, Singular.
They swim very fast.
She swims very fast.
English verbs only show agreement with number and person features of the nominal, but in some languages, the verb also agrees with gender of the nominal element.
Now let us look at the infinitival constructions, where the verb does not show any such agreement morphology.
They like to swim.
She likes to swim.
In these constructions, the verb (that comes with to) is an infinite verb (Note that the verb like is in its finite form in these sentences).
Modals or modal verbs, on the other hand, are like helping verbs or auxiliaries, which help define modality of the verb. In other words, they tell you about the ability, probability, etc. of the occurrence of the action expressed by the verb. In this way, they have some semantic content in them. Some modal verbs in English are would, should, might, will, etc.