It seems that there are two terms that could be broadly translated as "gratitude", which each express different aspects of the concept. הכרת הטוב denotes recognition, while הודאה has strong echoes of active admission. I know that both are brought up in the literature, even co-occurring in רש"י's comment on the birth and naming of יהודה. Does anyone explicitly contrast the two concepts of gratitude?
Hakaras haTov, although in a simple sense is an awareness like Yahu said, is a responsibility that manifests itself in an action or inaction, though not necessarily of hodaah.
An example off the top of my head is the Ramban on why Ammon was kicked out of the Kahal more than other nations who didn't take nourishment to the Bnei Yisrael. He answers that since Lot was saved by Avraham, Ammon had a responsibility to do good to the B'nei Yisrael, and they did the opposite.
Bringing nourishment is not "hodaah", any nation would have done so, and they did (Edom as well as Moav). But Hakaras haTov creates a higher responsibility of reciprocity toward those who benefited you. What actions or inactions will one take in recognition of being a beneficiary?
Hodaah is a verbal or physical acknowlegement of the benifit (like saying thank you). You can give hodaah to someone with good intentions even if you didn't actually have a benefit. In that case, you don't have a responsibility toward the benefactor, but you can be grateful for his intentions.
Rabbi Frand addresses the definition of each of these words in a different Dvar Torah.
On Parshat Shoftim he addresses the meaning of the words HaKaras HaTov which is generally translated as gratitude. He explains why they Talmud tells so many stories about Amoraim recusing themselves as viable judges just because they had received a favor from one of the litigants. "The major lesson that this passage of Talmud teaches is the concept of Hakaras HaTov [gratitude]. This Gemara teaches us how indebted each of these Amoraim felt to anyone who did them even the slightest favor." He seems to be indicating here that HaKaras HaTov is the indebtedness we feel and should show towards another. He explains that even acts that we have come to expect from another, like a spouse or an employee, should still induce a feeling of HaKaras HaTov.
On Parshat Tzav he addresses the meaning of Hoda'ah defining the word as both 'thanks' and 'admitting', giving credit for these definitions to Rav Hutner. We must admit we needed someone before we can thank them.
So if we take both his meanings and put them together it appears that we should have HaKaras HaTov toward everyone who helps us and serves us, whether we needed it or not and Hoda'ah is thanking someone for something we could not have done on our own.
Strict translation of Hakoras Hatov is gratitude / Hoda'a is praise.
When a person makes a Seudas Hoda'a for a Neis that happened to him he is praising Hashem for the Neis. Praising includes a proclamation of thanks. When a person appreciates what someone did to help him that is Hakoras HaTov. It does not necessitate a proclamation and can be done quietly in appreciation.
I noticed that there are two types of middos — intransitive and transitive. I mean those terms in the grammatical sense: Intransitive middos have no object, no specific target. Sadness is what I would call an intransitive middah, as it’s a state of mind rather than part of a relationship with another. Transitive ones express an attitude toward an object. Something close to a transitive version of sadness is disappointment; we don’t get sad at someone, but we can be disappointed in them. Admittedly the line gets blurry when one discusses a transitive middah but it’s about the relationship between me and myself. Is frustration with oneself really transitive? But we do feel a subtle difference when frustrated with ourselves and frustrated with “the universe”, with no specific target.
A second distinction I would like to draw before plunging into the topic of gratitude is that between perceptions and responses. We do not respond to the world, we respond to how we perceive the world. Like the famous picture that looks either like a beautiful young woman or her cronish mother, we choose how we see the world.
In the case of gratitude, there is certainly both an element of perception and of response.
The perception, in turn, also has two elements: Intransitive and transitive. In this section, I want to look at these two elements of the perception of gratitude.
I can feel grateful for having this apple. I have hakaras hatov, literally, a recognition of the good that is before me. I am happy because I have something to enjoy.
Then, I can turn it into a transitive feeling, going beyond being grateful for what I have, to look at who I am receiving from. I make a berakhah, I thank the One Who made this apple possible.
Within the hakaras hatov aspect of the perception, looking at what good we have in our lives, we can be aware of the same good thing in more or less detail. We can be grateful for the apple. Or, we can be grateful that Hashem made it possible for people to plant the tree, sell it to the wholesaler, provide the means for an open market in apples, all the many elements that go into my wife finding it in a store display, her loving me and wanting me to have something to enjoy and eat healthier, and her buying it for me. Notice how, by spelling out the detail, one realizes more fully the greatness of the tov. The closer we look, the larger it looms.
This is a primary lesson in meseches Berakhos. The gemara has discussions for dozens of pages over which berakhah to make on what food. Why? Does our thanks for a banana really depend on whether we thank Hashem calling it the fruit of a tree, or whether we recognize that a banana “tree” is a perennial, and therefore we should thank Him for “the fruit of the ground”? I think that’s just the point — the attention to detail is critical. Without it, one can not fully recognize the good Hashem does for us.
Along with recognizing the full extent of the good we receive is acknowledging that there is a provider.
So, to answer your question (finally):
Hakaras haTov is "recognition of the good" -- it's an intransitive perception. "It is good to be alive!"
Hoda'ah is "thankfulness" -- a transitive response. "Thank you G-d for giving me another day."
One of the things cancer taught me (BH and ba"h now 14 years in remission) was that there are two messages to waking up saying "Modeh Ani":
The explicit one -- thank G-d I woke up! (Hoda'ah)
The implicit one -- even "just" waking up is something huge (hakaras hatov). Which then gets me to the hoda'ah; it is worth thanking Hashem for.
(Source: translation of the words.)