How do I compare and contrast IDEA and Section 504?

You are correct that IDEA and 504 plans are similar, and you are not alone in finding the distinction sometimes hazy in practice. Both IDEA and 504 are designed to protect students from unwarranted discrimination or difficulty in receiving public education.
IDEA (Individuals with Disabilities Education Act) ensures that all students receive a Free and Public Education (FAPE). Prior to this act, it was common enough that students with mental or physical disabilities were denied access to public education. Under the IDEA plan, students are entitled to six accommodations, which I have tried to put in simple language as much as possible:
1. Individualized Education Program (IEP): schools tailor an educational plan to the student so that the individual ability and disability are accounted for. Students are asked to do as much as they can that is comparable to general education students but are not penalized for what they cannot do. Various methods to scaffold learning to assist student growth are reviewed frequently to ensure the plan grows with the student.
2. Free and Appropriate Public Education (FAPE): based on the individual disability, the school must provide free educational services that meet the educational goals of the disabled student as much as the general education student; the federal government funds about 40% of the extra cost of education a student with special needs. The family does not pay extra for the remainder of the costs of special education.
3. Least Restrictive Environment (LRE): to the greatest extent possible, the student with a IEP should be in the general education classroom with the rest of the student population. This ensured that schools could not place special needs students in one area and ignore the social and developmental growth that occurs when being in the general education classroom. In some instances, a student with a profound learning challenge may be pulled for one or more courses and taught be a special education teacher, but will return to other classroom when learning isn't impeded. A student with a profound reading issue may be taught outside a general education classroom but return for math if numeracy is on grade level.
If disciplinary action is required for students with either an IEP or a 504, certain legal rights apply. For instance, if a student is removed from school for more than 10 days due disciplinary actions, the school must then offer the learning plan where the student is (home or other facility). Misbehaviors caused by the disability (such as with Emotional Behavioral Disabilities) are not treated the same as that which is unrelated to a disability.
4. Appropriate Evaluation: students with an IEP must be tested regularly to ensure that the accommodations are appropriate to the needs
5. Parent and Teacher Participation: parents, teachers and special education staff meet regularly to review the student's IEP and ensure that current accommodations are appropriate and offering the least restrictive opportunity for student growth
6. Procedural Safeguards: these largely give parents considerable access to their students' regards and offer legal options should a problem arise.
Each of these, along with a high benchmark for confidentiality, creates a guidepost for schools and families to use to make sure that public schools are following best practices when educating a student with an IEP. The statutes set a high bar, the accommodations may be highly individualized, and the ramifications for failure to follow the law are steep.
Students with an IEP protected by IDEA will often have a medical diagnosis that falls within 13 well-defined categories of disability. With 504's the bar is a little different. Students may often receive a temporary 504 due to a shorter-term disability such as a concussion. Other students with identified but harder to diagnose challenges may receive a 504 if they have at least one impediment to learning but do not qualify for an IEP. Often, ADHD, chronic diseases such as diabetes or allergies, or other hidden conditions may fall under a 504. Fewer specialists weigh in on the need and accommodations in a 504, and generally the learning plan will address a smaller range of accommodations. For example a student with ADHD may be given more time to complete a general education test or may take the test in a different setting; a student with an IEP may be given a modified test. While schools are required to offer a 504 when appropriate and are penalized if they do not, no federal funding supports 504 plans.
In both cases, schools are required to maintain strict confidentiality within reason regarding the student's condition and accommodation. The goal is to guide the student with a disability through the public education system and to do as much as can be done to assist them in their college or career goals. Within any given public school classroom, a teacher may be offering several different accommodations based on the IEPs and the 504s in the class. Some of these are quite easy and subtle to implement (seating a student in the front of the class) while others are more onerous (writing different assessments) but the overall goal is to try to level the playing field as much as possible while keeping as many students as possible in the same classroom.