What is the difference between a missionary and an apologist when it comes to proselytizing for the Christian faith?

Similarities

• Both need to know the Bible very well as well as the chosen theological framework (plus the supporting documents & literature) that they want to promote and teach others
• Both need to be prepared to share the gospel in various settings: 1 minute elevator presentation, 15 minute house visit, 30 minute one-to-one conversation, 1 hour public sermon, 2 hour small group meeting, etc.
• Both need to know enough about the worldview, religion, language, and culture of the audience to mount a successful defense of Christianity
• Both need to be an exemplary Christ witness plus personal testimonial stories they can share in order to be more convincing (just like in other areas of life), although less necessary for an apologist who specializes in intellectual defense (such as scientist who can show the compatibility between faith and reason from within his field, such as biology)

Differences

• Missionaries are usually sent to serve long term (> 10 years) while apologists usually come by invitation to a conference, student club meetings, churches, etc.
• Missionaries need a supporting organization (like OMF) to help them with the long-term move, learning the foreign language, provide resources so they are acclimated with the receiving society's culture, help channeling the long term funding needed, become the liaison for visa purposes, provide mentorship with more experienced missionaries, etc. In contrast apologists are usually self supporting and they learn what they need from going to seminaries, self-study, etc.
• Missionaries often need to build an institution FROM SCRATCH: planting a church, seeding goodwill with influential members of the community, training local Christians to support the church, etc. We actually can learn a lot from how St. Paul established his churches and how he built on the existing Jewish financial network and Jewish synagogues. St. Paul was also an excellent apologist, being able to even create a brand new theology for gentiles, but he is more a special case of being gifted and supported by the Holy Spirit to be good at both. More often, apologists would PLUG INTO existing churches or parachurches already built by missionaries who came before them. Therefore, apologists come to help a new church becomes more mature. If apologists need to build an institution from scratch, they tend to build a seminary.
• As @Null pointed out, missionaries often encounter people who are ignorant about Christianity, so they need to be more versatile in using / creating evangelistic tools that are ATTRACTIVE and SUITABLE for the local culture, for example adapting BASIC good news in terms of local folk songs, dances, arts, poetry, etc. In other words, missionaries are specialists in "milk for spiritual babies". But as the new church grow, they also need to feed them with "solid food", which are also helpful to counter hostile people who are already knowledgeable about Christianity. (see this article for more of the difference of "milk" and "solid"). In contrast, apologists toolset can be more readily deployed in multiple culture with only minor adaptations.
• Missionaries are GENERALISTS who serve as PASTOR who need to minister to ALL AGES at ALL STAGES of spiritual development. Long term friendship with the congregation are critical, who then can introduce non-believers to come to be evangelized by the missionaries. But apologists can afford to be a SPECIALIST who focuses on a particular age range or a particular subject (science, theology, philosophy, liberal arts, etc.) and who don't need to form a long term relationship to be effective as most persuasion happens in the intellectual sphere.
• Missionaries are usually the ones who are given HEALING MIRACLES or other SPIRITUAL GIFTS especially if God (the giver) deem that the new Christians need miracles for a FINAL CONFIRMATION push after the gospel presentations. Instead, apologists work are usually either BEFORE the gospel presentation (to clear out false assumptions) or AFTER conversion in case intellectual doubts come, especially after their non-Christian friends challenge them. In both "before" and "after", the apologist's role is mostly intellectual and does not need a healing miracle to be effective.

Answering your questions

Do they tend to emphasize different skillsets?

As you can see from the differences above, YES, missionaries need cultural sensitivity skills, the ability to survive in a foreign country long term, and the experience to start a viable church from the ground up. On the other hand, apologists need to have a specialty so they need to be better educated in the subject matter.

Do they approach the task differently?

Yes. Missionaries are always long-term looking, and thus spend their energy more on what's more important in the long-term. They tend to avoid confrontation and they patiently wait for the right moment to challenge unbelief / wrong belief. They are like parents to children.

On the other hand, apologists are more "transactional" / short assignments, and thus spend their energy more on what's generating a quick result and maximum impact in the short-term. They are like guest speakers or school teachers.

Are there certain things that missionaries are usually better equipped to do than apologists (and vice versa), when it comes to proselytizing for the Christian faith?

Assuming that both missionaries and apologists are responding to their spiritual gifts and Holy Spirit calling, if the missionaries are sent to establish a new Christian presence (their usual role), they will be more equipped for proselytizing by being given the gifts essential in proselytizing: speaking well, miracles, and insights into people to lead them into conversion. They can then enlist the help of apologists to discuss specialized concern in the 3 way conversation, just like a general salesperson ropes in a field engineer into the sales meeting while the salesperson remains focused on the relationship aspect.

Notes

• From @curiousdannii:

  Not all missionaries are church planters - some are sent to simply live their lives in Gospel poor locations, taking up secular jobs in order to be witnesses to ordinary people. They'll still be trained in how to share the Gospel, but they don't necessarily have to have the spiritual gifts, training, or intention of being church planting pastors.