What is the difference between these two words?
promenas
promenante

Don't they both mean walking? In what situation do we use them.

Alkanadi:What is the difference between these two words?
promenas
promenante

Don't they both mean walking? In what situation do we use them.

The difference is the same like between:
walk(s)/is (am, are) walking
walking

Sentences:
He is walking around the room - Li promenas ĉirkaŭ la ĉambro
Walking around the room he saw him - Promenante ĉirkaŭ la ĉambro, li rimarkis lin

Alkanadi:What is the difference between these two words?
promenas
promenante

Don't they both mean walking? In what situation do we use them.

promenas (mi promenas...ili promenas)= Present Simple

promenante = present participle
walking down the street - promenante laŭ la strato

„I walk.“ ~ „Mi iras.“
„I am walking.“ ~ „Mi estas iranta.“

Fortunately the EO termination can tell us a lot.
"Promenas" ends with -as, which means that it is a verb in present tense, and it will function as a predicate in a sentence.
"Promenante" ends with -e, this makes it an adverb. It cannot be a predicate, it's function in the sentence is to describe a verb, adjective, some other adverb, or even another sentence.

„I walk.“ ~ „Mi iras.“
„I am walking.“ ~ „Mi estas iranta.“

Iri has a broader meaning than walk. Perhaps in German, gehen can be used for walk.

You can say mi estas marŝanta, mi estas promenanta for I am walking , but most often a simple form of the verb will be used.

How are you getting there? I am walking - Mi iros piede.

On a side note and maybe exceeding a bit the scope of the question:

Esperanto -ant- forms are always (and by definition of the Grammar of the Fundamento) present participles active.

In English, however, the ending -ing denotes two distinct forms: the present participle (active) and at the same time the gerund.

Present participle: I am swimming.
Gerund: I like swimming.

English present participles can be translated more or less directly to Esperanto present participles.
English gerunds are generally translated as Esperanto infinitives (-i), as Esperanto does not have gerunds.

So "I like swimming" or "Swimming is healthy" (both gerunds) would use in Esperanto the infinitive:
Gerund: Swimming is healthy. -> Naĝi estas sanfavore.
Gerund: I like swimming. -> Mi ŝatas naĝi.
Participle: He died swimming. -> Li mortis naĝante.

Alkanadi:What is the difference between these two words?
promenas
promenante

Don't they both mean walking? In what situation do we use them.

Promenas is the present tense of promeni, and promenante is the "active adverbial participle" of promeni. ( Like that clears anything up! ) Adverbial participles are words built on verbs that have both verb and adverb characteristics. Even that is too much detail. In practice these particular types of participles aren't terribly hard to understand or to use. The main use for -ante participles is to create concise subphrases in a single sentence, usually implying the preposition 'while'.

For instance, say you're writing a story:

Ĉapitro 1
Promenis Johano laŭ la strato.
John walked along the street.

Li vidis katon.
He saw a cat.

Good start, but maybe you want to tighten it up a bit. You could say instead:

Promenante, Johano vidis katon.
While walking, John saw a cat.

And that's all there really is to it. Promenante is just shorthand for Dum li promenis,.... It takes its tense and its subject from the main part of sentence.

You can also use -inte adverbial participles to mean 'something the subject did before the main verb of the sentence, and -onte something the subject intends to do after.

Participles in general are important, but there are *lots* of different kinds! They are a big subject (and a few were once quite contentious), and the only comprehensive treatments I've found of them online are written in Esperanto itself, (here's one) so it's best to not worry about them too much at the beginning unless you need to wax literary. In fact you see them mostly in works of literature, not as much in everyday talk. You can usually say the same thing using simpler words.

For further reading, check out the Esperanto version of Alice in Wonderland, it contains way more than its fair share of sentences with adverbial participles.