I suppose you are confused with the English infinitive to+verb. No. Infinitive is different in Greek.
Even it is different in Ancient Greek, that was similar with the English one in its use. It was a verbative form in -ειν (active) or -εσθαι (mediopassive voice). It vanished gradually in the middle ages and replaced by subjunctive να+verb.
The Modern Greek infinitive is the verbative form that is used with έχω to form present perfect, i.e. active: έχει λύσει (he/she/it has solved, and more), έχει γράψει (he/she/it has written), and passive: έχει λυθεί (he/she/it has been solved, and more), έχει γραφτεί/γραφεί (he/she/it has been written etc. So as no, it is not infinitive, no.
It's not that I didn't know about there being no inifinitive -- it's that I didn't recognise the "subjunctive" ending. Please see my other reply above.
No. THERE IS. But it is quite different in Modern Greek than in English, which is the same with the Ancient Greek one in its use. Example: έχω γράψει (active voice), έχω γραφτεί (passive voice) This γράψει and γραφτεί is the infinitive in Modern Greek, it is the same example as "I have written" and "written" would be the infinitive in English, but it isn't.
About Subjunctive in Modern Greek: It expresses will, wish,proposal or fear, so it is accompanied by these verbs . To understand if the verb is in Subjunctive we notice the if it is introduced with the conjunctions για να, όταν, εάν, άμα, μήπως, or the particles να, ας. Examples: για να έλθεις, όταν έλθεις, αν έλθεις, άμα έλθεις, Also ... να έλθεις, ας έλθεις.
The negation in Subjunctive is μη(ν), not δεν.
Notice that the Subjunctive is used in some forms of Imperative, but it is more mild, to express wish, but not order, να έλθεις, παρακαλώ, or ας έλθεις αύριο.
About the tenses in Subjunctive:
There are 3 tenses: Present, Past and Present Perfect.
Present: να έρχομαι * usually introduced with να in declension tables, even it isn't in all cases, see above. i.e. Θέλω να έρχομαι στο σπίτι σου
It is like the present in Definitive.
Past: να έλθω, να έλθεις, να έλθει, να έλθουμε, να έλθετε, να έλθουν. There is also the form να έρθω, ρ instead of λ.
More: να γράψω, να φάω, να πιώ, να πάω, να διαβάσω, να βρω etc, etc. I think it's the most difficult case, but you have to remember the first person only, and only for irregular verbs, the other are similar.
Present Perfect: να έχω έλθει. It is similar like Definite in the same tense.
Some Grammar books use the terms Εξακολουθητική (Continuous), Συνοπτική (Summary, concise), Συντελεσμένη (Perfect) Υποτακτική (Subjunctive), for the above tenses. These terms are used in the school grammar for Greek children of highschool. The above ones are older terms.
What makes the Subjunctive difficult for the English native speakers? It is fact that it is under extinction in English, I think, that is not the case for other languages. My personal view is that it is easier than other languages which use so many different suffixes that can make even a native Greek speaker to scream :)
I really don't think you can describe γράψει as an infinitive. An infinitive in all the other languages I know is both personless and tenseless. γράψει clearly has an associated tense.
My problem with the subjunctive here probably has more to do with how I am learning Greek using the Michel Thomas (MT) method (all rights reserved!) and Duolingo but with no grammar books. On the MT audio it is said that you use the subjunctive after πρεπει να, θελω να, μπορω να and θα. The only form of μιλαω we have had in the subjunctive is μιλησω i.e. with the σ and this was in a present tense situation! I think the example given is something like: θελω να μιλησω στα ελληνικα με σας. Is this incorrect?
I had come to the conclusion, both from Duolingo and MT, that αρεσει να was followed by the indicative and therefore expected the verb in the example here to be μιλαει and upon being told μιλα was the subjunctive I wondered why the correct answer was not μιλησει.
To answer in brief to your example: Is there any difference between:
Θέλω να μιλάω ελληνικά μαζί σας and
Θέλω να μιλήσω ελληνικά μαζί σας ?
Yes, there is.
In the meaning and Grammar:
For 1) : I want to speak Greek with you (it is implied from now on or whenever we speak each other), that is Continuous (Subjunctive of Present).
For 2): I want to speak Greek with you. (it is implied that this very moment, because .... , but not exclusively in Greek every time we speak each other). That is Περιληπτική Υποτακτική (Summary Subjunctive), Subjunctive of Past.
The form Θέλω να έχω μιλήσει Ελληνικά μαζί σας, is not used, Subjunctive of Present Perfect, it is meaningless. Instead the sentence Θα ήθελα να έχω μιλήσει is quite usual.
It is also used the form: Θα ήθελα να είχα μιλήσει, but it is not grammatically correct, there is no Subjunctive of Past Perfect. There had been a debate about this level in the past that exists in the Greek tree : https://www.duolingo.com/comment/17787787/I-could-have-asked-this . It is used as a redundant form to emphasize that the action had finished before our wish, but the linguists reject it.
The above definition of the infinitive is from grammar books, specially the book taught to Greek students of Junior high school, it says in Greek: Το απαρέμφατο είναι άκλιτος τύπος του ρήματος και χρησιμοποιείται για τον σχηματισμό ορισμένων χρόνων στην ενεργητική και την παθητική φωνή, π.χ. έχω δέσει, είχε δεθεί. My translation, sorry if it's not good English: The infinitive is a verbal form without declensions and it is used to make some tenses in the active and passive voice i.e. έχω δέσει, έχω δεθεί. I have tied, I am tied up. http://ebooks.edu.gr/modules/ebook/show.php/DSGYM-A112/621/4006,17975/
About the Subjunctive. I wrote all I know and I used this Grammar book and the classical book by Manolis Triantafyllidis.
Well, the form introduced with θα is Future in Definitive. Future most of the time is like this. Θα+verbal form. There are other cases too, but generally it is so.
The form μιλήσω is one of the tenses in Subjunctive. It is for once, not Continuous, when it is used μιλάω, μιλώ. I wrote about it previously. Να μιλήσω is considered as a Subjunctive of Past tense or a Περιληπτική Υποτακτική, once. Να μιλάω, να μιλώ is NOT a present tense of Indicative. It is a continuous tense, that is the same though in its form. But all Grammar books consider it as separate tense in the conjugation table. The same with να έχω μιλήσει, it is not Indicative, even it is the same with the Present Perfect of Indicative. This conjunctions I mentioned before or να make them Subjunctive.
About Subjunctive, and all moods of Modern Greek take a look here, it is a good treatise on the subject: https://www.greekgrammar.eu/moods.php
Grammar has many approaches, as it happens to all taxonomies.
There's no difference in meaning. You'll find many words in Greek that have various spellings, and /or pronunciations that are equally correct. It's just a matter of personal habit, sometimes regional preferences. A lot has to do with the fact that Greek is so old and has gone through so many changes that some of them overlap.
Thank you - nice explanation - I've always wondered about that one with -αω verbs.