Lets and allow change sentences, why?

Just because two words have similar meanings does not mean they are interchangeable—almost no two words are truly interchangeable, on several levels. Particularly here, two words being similar in meaning does not mean their grammar needs to be similar, either. Every verb has its own conventions as to which complements are accepted or not.

When using a verb form to indicate an activity which is being made possible, allow takes a to-infinitive as an object complement or a gerund as a direct object, e.g.

This allows you to jump

This allows jumping

This is also the case with authorize, license, enable, approve, release, sanction, okay, suffer, bless, warrant, and green-light among some other verbs of permission, approval, or empowerment.

Let only accepts a bare infinitive as an object complement for the same, and is the only common verb of permission, approval, etc. I can think of which does.