I agree with @Erik, these instruments are essentially equivalent, but I am going to add a few more ideas about this topic.
In order to provide more context, it is necessary to have an authoritative definition for both terms. So, here are some excerpts from the relevant paper entitled Relational Completeness of Data Base Sublanguages, which was writen by the originator of the relational model, Dr. E. F. Codd, and published in 1972:
Relational algebra. The primary purpose of this algebra is to provide a collection of operations on relations of all degrees (not necessarily binary) suitable for selecting data from a relational data base.
Relational calculus. Having defined a relational algebra, we now consider an applied predicate calculus which may also be used in the formulation of queries on any data base consisting of a finite collection of relations in simple normal form.
In this way, one can see that they serve the same purpose, since they are two different mathematical means to express a group of operations considered valid within the relational framework.
Comparison of both approaches
In the same paper, Codd contrasts both methods in order to point out some important aspects in which each one stands out:
One advantage that might be claimed for the algebraic approach is its freedom from quantifiers. Howerver, the calculus appears to be superior to algebra in four respects.
- Ease of augmentation
- Scope for search optimization
- Authorization capability
- Closeness to Natural Language
So, these aspects should be taken into account by organizations that aim to build their own languages or database platforms.
I do not know if there are contemporary successfully commercial RDBMS that provide query languages purely based on relational calculus, but I deem that its practical worth lies in the fact that it offers a theoretically grounded alternative for vendors who plan to design and implement a new valid relational language, or extend an existing one without departing from proven and accepted practices. In this respect, it can be useful to think of relational calculus as a blueprint that such vendors can follow in order to produce a relational language.
Since the market is largely dominated by SQL platforms, millions of users around the world are already familiar with several of its particular dialects, and billions of dollars are invested in such platforms, it would be very interesting to determine whether creating new relational lenguages would be profitable, but if so, relational calculus could always play an important role.
Of course, such projects can be much more feasible in the academic field, so there is more practical value regarding this tool.