What is the difference in pronunciation and meaning between these syllables?
Theoretically, 왜 is a sound that combines ㅜ and ㅐ, while 웨 combines ㅜ and ㅔ. However, it is realistically difficult to differentiate the two pronunciations. Both of them sounds [we]. (It is not [wee].)
You can think 왜, 웨, 외 those three sound the same.
However, the exact sounds are a bit different.
ㅗ+ㅐ / ㅜ+ㅔ / ㅗ+ㅣ
oe / ue / oi
And in general, Koreans pronounce them the same[we].
The standard pronunciation still differentiates 왜 /wε/ from 웨 /we/. /ε/ is the open-mid front unrounded vowel, and /e/ is the close-mid front unrounded vowel. If you are not familiar with IPA symbols, checkout the links to see if you speak any of the languages that distinguish the two.
However, the distinction is completely gone in the modern Seoul dialect and also in most regional dialects. Only the elderly population in certain regions articulate them differently. Today, the two phonemes are merged into one, which is often written /E/ by Korean phonologists. It is not a standard symbol, but one specific to Korean phonology, referring to the vowel ranging from /e/ to /ε/. Accordingly, 왜 /wε/ and 웨 /we/ have merged to /wE/.
If you are a learner and don't distinguish the pairs of phonemes /e/ and /ε/ or /we/ and /wε/, (the Wikipedia pages I linked above have audio samples, so give it a try), it is totally fine to pronounce them identically. If you distinguish the phonemes, feel free to pronounce them the standard way if you want to. It won't mean any difference to most native Korean speakers, but may help you memorize some spellings.
Note that another vowel, ㅚ also underwent a sound shift to /wE/. Its standard pronunciation allows both /ø/ and /we/, but you will hardly hear it pronounced /ø/ in most regions. Now, the three vowels ㅚ ㅙ and ㅞ don't sound different to a contemporary Seoul speaker.
As you also asked of meanings, 왜 is most likely the interrogative "why", but may also refer to Japan in a historical or derogatory sense.
The syllable 웨 does not appear in any native Korean word (at least in the standard spelling), and if you see one, it is most likely a misspelling of 왜/외 or a transcription of a foreign sound, for example, 웨스턴 /wesɯtʌn/ from English western or 웨이브 /weibɯ/ from wave.