We usually find this technology referred to indistinctly by these two names, which can lead to confusion on the part of the non-expert public. And the truth is that even many professionals, myself included, use them as synonyms.
Two terms for the same process?
Is there a difference between the two designations?
We will see what the standardisation norms say about it, its mention in scientific articles and what the general public looks for on the internet in order to get an idea of what we are talking about.
What the standard says
To find the answer to these questions, we must consult the bibliography available from the world authority on standardisation, the International Organization for Standardization (ISO) based in Geneva, Switzerland. We take advantage of the fact that just this November 2021 the update of the standard that designates the vocabulary and principles of additive manufacturing, ISO/ASTM 52900, has been published. 
The standard defines the term additive manufacturing as follows:
“Process of joining materials to make from 3D model data, usually layer upon layer, as opposed to subtractive manufacturing and formative manufacturing methodologies”
In the case of 3D printing, the ISO defines it this way:
“Fabrication of objects through the deposition of a material using a print head, nozzle or another printer technology”
Note 1 to entry: This term is often used in a non-technical context synonymously with additive manufacturing and, in these cases, typically associated with machines used for non-industrial purposes including personal use.
The two definitions are actually very close, and in the case of 3D printing, the first thing that might come to mind is a desktop FDM printer. It is Note 1 associated with that definition that gives us a clue as to the difference between the terms. It states that, although often used synonymously, the title “3D printing” is associated with non-industrial activities.
In fact, in the new standard published in 2021, this is one of the paragraphs that has undergone the greatest modification, as in the old 2015 edition it spoke directly of “machines low end in price or overall capacities”.
What science says
Another source of value and rigour in clarifying not only the differences but the weight of each term in the technology is the way it is referred to in the scientific literature.
There is a similar level of confusion and lack of definition, to the point that even the term “rapid prototyping” has been used as a synonym for additive manufacturing processes, particularly with SLA technology. 
A good indicator are the keywords, the words that the author has decided to highlight for the article to be found. In the research by W. Bai et al, we see that it is the term “3D printing” that has established itself as the most widely used keyword, followed by “additive manufacturing”. 
Indeed, if we search within Springer, a scientific publisher, we see that 3D printing gives 118.171 results against 105.708 for additive manufacturing. 
There is however a journal in Elsevier dedicated to the field called Additive Manufacturing Journal.  The curious thing is that when a journal refers to a particular area of knowledge, it refers to it as “3D printing”. , 
What Google says
Internet searches give us valuable information about people’s trends and knowledge. However, these data are unbiased, as they refer to what anyone has typed into the browser, without distinguishing between population profiles.
The results by keyword for the average number of Google searches per month in the English language for the whole world are as follows: 
We see that the number for 3D printing is about 7 times higher than for additive manufacturing, implying that it is a term more familiar to a higher percentage of the population. The same is true for the French and Spanish languages (see translation), with the search volume being considerably higher in English.
What the author says
From my humble point of view, both terms are valid to refer to this technology, as a general approach of the technique or just referring the manufacturing process. They are and will continue to be used synonymously and this is not necessarily a mistake. Language has an influence, as it is easier to say “I need 10 parts to be printed” than if we have to refer to the same thing by talking about additive manufacturing.
It is in many cases a question of marketing and positioning, as for example, adding 3D to a particular brand name can give a quick indication that it works in this field. Others, such as this blog, have preferred to use the term additive manufacturing to give a more industrial approach to the content published.
Cover image, collage by the author, sources:
P. Calves, Metal Binder Jetting une opportunité pour la production de petits composants complexes en petite et moyenne série, Traitements & Matériaux, 452, 50-56, Mai-Juin 2018
J.R. Tumbleston et al, Continuous liquid interface production of 3D objects, Science, 347, 1349-1352, March 2015
 ISO/ASTM 52900:2021 Additive manufacturing — General principles — Fundamentals and vocabulary
 Web ISO/ASTM 52900. Accessed 1/12/2021
 K. V. Wong & A. Hernandez, A Review of Additive Manufacturing, ISRN Mechanical Engineering, Volume 2012, Article ID 208760, Jun 2012
 W.Bai, H. Fang, Y. Wang, Q. Zeng, G. Hu, G. Bao & Y. Wan, Academic Insights and Perspectives in 3D Printing: A Bibliometric Review, Appl. Sci. 11, 8298, Sept 2021
 Springer Link. Accessed 1/12/2021
 Additive Manufacturing Journal, Elsevier
 3D Printing in Medicine, Springer
 Polymers for 3D Printing, Elsevier
 Keyword analysis by Presentia Comunicación Web 1/12/2021