If I was to translate de as "and" as my grammar says I can.....how would that differ from kai?
Look at Morph Search results for lemma:δέ OR lemma:καί in Matthew 1, ESV. One stands out as a pure conjunction and the other stands out as a marker of addition.
Not much, if any. It might be softer than kai, but I wouldn't hinge any theological distinction on it.
When it comes to "but" de tends to be a soft contrast whereas alla is a strong contrast.
EDIT: Consider BDAG's entry on de's contrastive side
① a marker connecting a series of closely related data or lines of narrative, and, as for. Freq. used in lists of similar things, with a slight call of attention to the singularity of each item (cp. Hom., Il. 3, 144–48).—In tightly knit lists Mt 1:2–16; 2 Pt 1:5–8; relating one teaching to another (in this respect δέ is similar to the use in 2) Mt 5:31; 6:16; Ro 14:1; 1 Cor 7:1; 8:1; 12:1; 15:1; 16:1. Freq. w. the art. in narrative to mark change in the dramatis personae, e.g. Mt 14:17f, Mk 14:31.William Arndt et al., A Greek-English Lexicon of the New Testament and Other Early Christian Literature (3rd ed.; Chicago: University of Chicago Press, 2000), 213.
http://homepages.nyu.edu/~dhl271/Site/NYU%20Linguistics_files/Lassiter%20Theocritus%20HWPL.pdf
will tell you more than you want to know about de which is a particle of emphasis. kai I believe is a simple conjunction.
Here is a very brief, and incomplete excerpt from Runge, S. E. (2010). A Discourse Grammar of the Greek New Testament: A Practical Introduction for Teaching and Exegesis (42). Bellingham, WA: Logos Research Systems, Inc.
One of the significant mismatches between English and Greek conjunctions is clearly seen in the different senses that are ascribed to καί. The primary senses are connective (“and”) and adversative (“but”). These two English conjunctions, however, mark an inherent semantic quality that is not marked by either καί or δέ. ... This semantic quality that distinguishes and from but is not marked by καί. It may or may not be present. The same is true with δέ.
Καί is a coordinating conjunction that may join individual words, phrases, clauses or paragraphs.
In most narrative contexts, καί functions as the default means of coordination. The use of καί constrains the connected element to be closely associated with what comes before, regardless of whether there is semantic continuity or not. The implication is that the elements joined by καί are of equal status.
BDAG say that δέ is “used to connect one clause to another, either to express contrast or simple continuation. When it is felt that there is some contrast betw. clauses—though the contrast is oft. scarcely discernible—the most common translation is ‘but’. When a simple connective is desired, without contrast being clearly implied, ‘and’ will suffice, and in certain occurrences the marker may be left untranslated”. However, as with καί, the connective δέ does not mark the presence of semantic discontinuity, as BDAG claim.
Grammarians have worked diligently to make καί correspond to and, and δέ to but, which has lead to great confusion regarding the unique grammatical role that each plays.
There is no simple answer to your question. To get a satisfactory grasp of all these prepositions will require a great deal of study and effort on your part.
Whilst I agree with Alex and Martha and also agree with Kevin it does not mean that I'm using kai for and in this sentence as one of them comes from de[:)]
If you have "A Discourse Grammar of the Greek New Testament", you might try reading what Steven Runge has to say concerning the use of these conjunctions. They are not the same, although at times the differences can be subtle. "And" and "but" translations are alright for Greek 101, but when you examine a quality lexicon, you will see that this does not cover all the connotations of these conjunctions.
EDIT: I had not seen Alex's response when I composed the above. I agree with all that he said, especially
Except, they are conjunctions rather than prepositions.
