the difference between “really” and “very”
Solution 1:
I don't agree with that distinction. If you look up really and very, they both can be used as intensifiers. (In your example sentence, they intensify the word "hot.") I don't believe one is more "personal" than the other.
I do agree, however, with Theodore Bernstein's entry on the word very in his book The Careful Writer:
...An aside on the word very... Inexperienced writers tend to use the word too much. Often its use is self-defeating; the writer intends to intensify what he is saying, but instead weakens it. He may write, "Hemingway's prose is very lean and very strong," not realizing that he would express his thought more forcefully if he wrote, "Hemingway's prose is lean and strong." If the word very seems to be necessary to strengthen what has been written, the writer should re-examine his original selection of words. Strong words usually need no such prop."
I believe the same is true for the word really.
Solution 2:
They can both be used to intensify, but they're not always interchangeable. I can say:
"I really like JLG's answer."
but I wouldn't say,
"I very like JLG's answer."
Correspondingly, I could say,
"I like JLG's answer very much."
but I probably wouldn't say,
"I like JLG's answer really much."
(That's just an example, even though I really do think it's a very good answer.)
That parenthetical statement brings up one other distinction: really can mean truly, as well as very. So, if I say,
"I'm really proud of my daughters."
that could mean two different things. It might mean, quite simply:
"I'm very proud of my daughters."
or, it might mean,
"No, I'm serious – I truly am proud of my daughters."
(The latter might be used in the case where, say, I had told someone that I was proud of my daughters, and they seemed somewhat incredulous, unaware that I even had a family.)
Solution 3:
Really and very are interchangeable when modifying an adverb or adjective.
For example:
He eats very quickly. (correct)
He eats really quickly. (correct)
Very/really modify the adverb quickly.
It is very hot today. (correct)
It is really hot today (correct)
Very/really modify the adjective hot.
We do not use very to modify a verb.
For example:
She really likes ice cream. (correct)
? She very likes ice cream. (incorrect)
Really modifies the verb like. You can not use very!
Solution 4:
As others have said, very and really can both be used to intensify. A main difference is that we don't use very to modify verbs.
He very runs quickly. (incorrect)
He really runs quickly. (correct--really is used with verbs)
He runs very quickly. (modifies the adverb quickly)
He runs really quickly. (also correct--this is where very and really are interchangeable)
Solution 5:
According to the OED, really used to be somewhat more than very, but it now also employed as an alternative intensifier:
b. Truly, indeed; positively. In later use also as an intensifier: very, thoroughly.
I believe this connection with something authentic (as in truly) is still present in common speech, by using really rather than very in saying, e.g, a really strong man you imply that your emphasis is not comparative but that the strength of that man was not to be compared with ordinary men's.
The association with authenticity is indeed still present in another usage of 'really':
c. really and truly: authentically, genuinely; honestly. To know what humans are really and truly capable of, the good and the bad, we need...
But we tend to over-use intensifiers, and that lowers their market value, as was already noted in 1742, according to OED again:
H. Fielding Joseph Andrews I. ii. xiii. 257 The Word really and truly signifies no more at this day.