Detecting protein and elevated levels of amino acids in the urine is generally a bad thing and is usually indicative of an underlying disease.
Kidneys are supposed to filter out waste products and retain metabolites that your body needs. There are many possible diseases that can cause amino acids to be detected in the urine, some as innocuous as overexertion or exposure to temperature extremes, and some as serious as Hodgkin's lymphoma, Amyloidosis, Malaria, Lupus, etc. For a more complete list see this Mayo Clinic page.
As for the blood, there are certain normal levels of amino acids found naturally in the blood. The reason for testing these levels is that certain imbalances can be indicative of different medical conditions. A doctor can use these levels to help to identify a problem that may not be noticed or may use it to diagnose a disease that you are seeking treatment for. This MedlinePlus page, has information.
Testing of Blood and Urine amino acids can and will be done in conjunction with one another in order to help narrow down the possible diagnoses that are likely.
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Edit to address comments
"Also, based on your answer and the Mayo Clinic link, what I understand is in a healthy individual - testing amino acids from urine might give false positives?"
I think that your concept of false positive here is incorrect. A false positive for this test would be indicating higher levels of amino acid in a sample than the sensitivity and standard error for the test would allow for.
That there is a laundry list of reasons why one might find abnormal levels of amino acids only goes to the fact that the kidneys "see" the results of several metabolic processes (normal or abnormal) and urinalysis is only a single datapoint from a spectrum of data for a clinician to begin to recognize a pattern and formulate a diagnosis.
As for greater accuracy, when comparing "Urine Random" to "Urine 24 hour" I would say that the "Urine 24 hour" test is more accurate strictly from a perspective of statistics. The larger the sample size for any test, the greater the level of confidence that the results observed are not due to random chance, error, or outliers in the data.
But this is the same concept for a political poll, marketing research analysis, or medicine; larger sample size gives you a clearer picture, but it does so at an increased cost. So the single screen is there just to look for issues, possibly based on a complaint that brought you into the doctor's office, or a way of monitoring a previously diagnosed condition and/or treatment, while the time courses are done to provide a clearer understanding of what is representative for the levels in that individual.
The single test has a number of reasons why it might be anomalous. Maybe the elevator from the underground parking garage was out of order and you had to walk up several flights of stairs to the office where they were collecting the sample, so it would show an elevated result due to exertion. Or maybe your drank a liter of water before the exam and your urine is dilute, causing a false negative. So to mitigate those anomalies, you run the time course, but usually only if there is a problem suspected because of the fact that it is an increased and unnecessary cost if all other indications are that the person is healthy or the result is easily explainable due to other data collected.
"I also understand that at times they would be used in conjunction - which seems logical. What I am trying to understand is which would give more accurate 'picture'. If one is not more accurate than the other, then what would be the difference between the 'picture' acquired from a blood test and the urine test."
Each test has its own parameters and give you an independent picture, that can be used to put together a more complete whole. Their accuracy will be based on their own sensitivities and built in standard errors and one will not be more accurate than the other because comparing free amino acids in the blood versus in the urine are two different pictures with different normal outcomes.
More samples, i.e. "Urine 24 Hours" or multiple blood draws over a time course is really the only way that you will improve on the accuracy of the individual tests, for the reasons I gave above.
They also have different baselines. Blood you expect to find free amino acids and there are ranges of normal results that you would expect to find in the blood. Here, too much or too little amino acid would be an indicator of a possible problem. For urine on the other hand, one would not expect to see elevated levels of amino acids, so here, a test that indicates elevated levels of amino acids indicates that more follow up is needed.
Think of it in terms of an analogy about astronomical observation. The electromagnetic spectrum is far broader than any one instrument can detect, so astronomers/astrophysicists use different sensors; visible light, radio waves, infrared, x-ray, etc. telescopes to detect many of these different pieces of data coming from the same object to gain a better understanding of what they are studying as a whole. Each gives slightly (or very) different views of the object, all of which contribute to a more complete picture of the object as a whole.