What is the difference between 募集｛ぼしゅう｝ vs 採用｛さいよう｝?
Looking at the 漢字 is quite informative.
You can see that 募集 is made of 募+集. Interestingly, both can be used to build verbs: 募る (to recruit) and 集まる (to gather). So 募集 means "gathering applicants" (or something like that). 募集 is seen a lot in the form of 募集中 (We are recruiting now). Note: The use of 募集 may not be limited to work: even if it can feel a little desperate 友達募集中 would mean that you are looking for friends.
Now, if we look at 採用, again we can decompose it into 採｛と｝る (to pick) and 用いる. From this observation we can see that 採用 is more about picking the right candidate than anything else. That's why there are 採用試験 but no 募集試験.
In conclusion, both mean to hire, but 募集 emphasizes the process of gathering applicants while 採用 emphasizes the recruiting process (that may be competitive).
Looking at this question here it seems as though 募集 refers to the planning of recruitment whereas 採用 refers to recruitment where the person/group decision has been made.
If you look a bit into the kanji, 募集 contains 集 which means to gather, bring together whereas 採用 contains 用 which means to use.
Also, 募集 usually refers to recruitment whereas 採用 refers to employ/hire.