Marian, you hit a great target! This is one of the most interesting verses in the Bible, for which we could be discussing endlessly.
You ask if there is a difference between εἶναι and γίγνεσθαι. These are foundational terms in philosophy. Εἶναι is identified with permanent/immutable existence, γίγνεσθαι with change. Εἶναι belongs to the Creator, γίγνεσθαι to the creature. Γίγνεσθαι is coming-to-be, is be-coming, not being. Yet, think how strange it is, when εἶναι gets half of its tenses from γίγνεσθαι!
As you remember, εἶναι has proper forms only in the present, imperfect and future tenses (εἰμί, ἦν, ἔσομαι), while in the aorist, present perfect and pluperfect it borrows the types of γίγνεσθαι (ἐγενόμην, γέγονα, ἐγεγόνειν)! This most foundational verb in any language, the verb of being, the verb of expressing the existence of anything, upon which the whole language is based, this verb in Greek is double-sided, if I can call it thus, having one side in being and the other side in becoming. In the grounds of Greek, these two usually juxtaposed verbs, defining metaphysics from antiquity until now, the verbs εἶναι and γίγνεσθαι, the eternal opposites!, are united – and more than this: while γίγνεσθαι can exist without εἶναι (it has all its tenses from itself and it can rely on itself), εἶναι can not exist fully without γίγνεσθαι! As John says in the third chapter, "God loved the world so much, that he gave his only begotten Son"...
On the use of Greek tenses you can have a look at Elpenor's Greek lessons, there is no need to repeat myself here, especially when there are other things we have to notice in this passage. Let us give the quote first:
Ὁ ὀπίσω μου ἐρχόμενος ἔμπροσθέν μου γέγονεν, ὅτι πρῶτός μου ἦν (John 1.15: The one after me coming, before me has become, for first than me he was).
The author of this text, who is known by the name of John according to the Christian churches which recognised in him a wisdom so great, that it deserves the authorship of the beloved disciple, the author of this text shows a great understanding of language. Perhaps John wrote this phrase spontaneously, as it was revealed in his poetical/philosophical mind, but ourselves can surely examine it and see what his faith and love were able to achieve through language.
There exists in this short phrase, there is described/expressed, the whole of time and of eternity as the time and the eternity of Christ. Notice the two parallel lines, the one with adverbs (after, before, first), the other with verbs (coming, has become, was): John "fills" with Christ the whole of time and all the relations that can exist in time (after, before, coming, has become) while placing Him also in the origin of time, beyond time (first, was). Now let's see how the tenses co-operate in John's text, something the translator that you mention failed to see. But if we miss this and translate the perfect as a present, miss also the importance of the participle, etc, we just destroy the meaning of John.
In the verb-line we have a present (ἐρχόμενος, coming), a perfect (γέγονεν, has become) and an imperfect (ἦν, was). The present is expressed by a participle. Why? John could have said "He, who comes after me," etc. (Οὗτος, ὃς ὀπίσω μου ἔρχεται, etc, or just ὃς ὀπίσω μου ἔρχεται, etc). Ὁ ἐρχόμενος transforms the first of the three parts of the phrase, the part that introduces the person about whom we speak, i.e. the Christ, into a union of two words, making of the action an epithet, infusing into Christ as His essential characteristic that He comes.
If we say ὃς ἔρχεται, we just describe one among many actions (ἔρχεται, γέγονε, ἦν, etc), while the way John said it, the Coming of Christ is distinguished from the actions that follow. It is not the Christ, who also comes, has become and was: it is the-Coming-Christ, who also has become and was.
"The coming", means coming after us, following us, serving us. This is the only pure present in the phrase, and it is this present that is escorted by the perfect and transcends the imperfect. The imperfect is His being in the Beginning (He was first) and the perfect is His importance for our salvation due to His being God in the Beginning - but even this importance does not stop Him from being the-coming-Christ, on the contrary: if He were not the coming, the serving, Christ, He would not have given Himself the chance to become also in front of us and lead.
Ἐρχόμενος is a participle, it means participation. The other verbs (γέγονεν, ἦν) describe an action or condition that is subjected to the person who is in that condition, while ἐρχόμενος means a person's participation in the coming. The Coming, this way is the principal concept, the principal action and condition. To be a little provocative I would say that if He was not coming, He wouldn't be the Christ. He is the Christ because He comes, He follows, He serves and saves. From this point, God is not self-sufficient, God's love is greater than His omnipotence, as John says, "God loved the world so much, that he gave his only begotten Son"...
Therefore, a translator should express the coming using a participle, he should by all means avoid putting three verbs in a row, and he should also use the perfect to describe the becoming that comes out of coming, and the imperfect that describes His being beyond time.