Use both! Illustrator is great for creating your logo/brand or decorative components. Then drop them all into InDesign, which is made for page layouts/publishing.
A few benefits on using InDesign for your the format of your resume, or text-heavy content, include:
- You can set up master pages so your header/footer is the same on every document easily (so your cover letter/reference/resume/thank you note all match for brand consistency). Can do the same with setting character styles.
- You can easily add/duplicate pages to adjust content per job application (keeping a record of what you sent- limitless pages, easy to navigate)
Hope this helps!