primary 第一位的，主要的，初步的，初级的，原来的，根源的
elementary 初步的，基本的，（化）元素的，自然力的


用法详解如下：

pri‧ma‧ry
1 [usually before noun] most important [= main]:
Our primary concern is to provide the refugees with food and healthcare.
Many of the villagers rely on fishing as their primary source of income.
primary purpose/aim/objective
Their primary objective is to make money.
Personal safety is of primary importance.
2 [only before noun]SES especially British English relating to the education of children between five and eleven years old [= elementary AmE; ↪ secondary]:
a primary teacher
primary education
teaching at primary level
3 happening or developing before other things:
a primary tumour
Counselling was given as a primary therapy for depression.


el‧e‧men‧ta‧ry
1 simple or basic:
the elementary principles of justice and democracy
You've made a very elementary mistake.
2 [only before noun] concerning the first and easiest part of a subject [↪ intermediate, advanced]:
I'm only familiar with the subject at an elementary level.
I know a little elementary science.
3 [only before noun] American EnglishSES relating to elementary school [= primary British English]
elementary education

They are similar in meaning.    Primary many times is associated with the first step , or the beginning.


1. Elementary, primary, and rudimentary refer to what is basic and fundamental. Elementary refers to the
introductory, simple, easy facts or parts of a subject that must necessarily be learned first in order to
understand succeeding ones: elementary arithmetic. Primary may mean much the same as elementary;
however, it usually emphasizes the idea of what comes first even more than that of simplicity: primary steps. Rudimentary applies to what is undeveloped or imperfect: a rudimentary form of government.

Primary may mean much the same as elementary; however, it usually emphasizes the idea of what
comes first even more than that of simplicity: primary steps. Rudimentary applies to what is
undeveloped or imperfect: a rudimentary form of government.

Source:
http://dictionary.reference.com/browse/elementary?db=luna