Just for reference, according to an online Chinese dictionary 烟 appears to be considered the simplified variant of 煙.
I don't think that the right-hand components have anything to do with the meanings here. Both of these look to have right-side phonetic components relating to their on-reading of エン. Other kanji using 因 have readings of イン or エン, as do other kanji using 垔.
形声文字{けいせいもじ} is the term for these sort of kanji, which have a meaning component 意符{いふ}, and a sound component 音符{おんぷ}. It only works for some on-readings, but it's common enough to take note of. 張・帳(長・チョウ) or 胴・銅(同・ドウ) are examples. An example of one which is not a left-right split would be 草 which has the "grass" radical on top and the same on-reading (ソウ) as 早 does.
So there is probably no meaning difference (or at least, there is none in the modern language). However, there would be a stylistic difference in the choice of a kanji variant over what is now considered the standard form, and there may be cases such as names where the use of one or the other is now fixed and they are not interchangeable.
e.g. here there is an example of a waterfall named 飛烟の瀧 (picture of the sign included, they do put furigana on it). I believe this would be a case where using 煙 would be incorrect.
ETA after seeing your comment:
I have seen suggestions that sometimes the original choice of phonetic component could be related to the meaning, for example that 清 uses 青 not just for the sound, but because the meaning is related. (This theory is apparently called 右文説). You probably need to be into old Chinese to really dig into this.
The Wiktionary entry on 煙 suggests that the 垔 side may have an origin from a combination of 西 (here being equivalent to "window") and 土 (here being equivalent to "hearth"), so representing something like window/opening above a hearth where smoke emerges from a fire. It also says that 烟 is a shorthand form that has been around for a while.
If that's correct, then just as you sometimes see kanji in compounds swapped for another kanji with the same reading which doesn't necessarily have the same meaning (for example 綺麗 being written 奇麗), 因 could have simply been swapped in for 垔 in this kanji as slightly quicker to write. Or there might have originally (in Chinese) have been some subtle difference in meaning which has since been lost.