There's no single reason why Caesar and Pompey drifted apart after Caesar's consulship. A large number of possible contributors has been put forth over the years, including Crassus' death, Julia's death, Clodius' betrayal of Pompey, Pompey's jealousy of Caesar's accomplishments in Gaul, etc. At one time it was thought among certain circles that Crassus' death was the significant destabilizing factor between Caesar and Pompey, but most scholars find that pretty weak now, and have for some time. There are really only a handful of reasons to cast Crassus as some sort of stabilizing influence on Caesar and Pompey, and none of them are strong. Chief among them is probably that within a year of Crassus' death Pompey had marched his soldiers into the Forum and Pompey had been appointed sole consul by the senate. But that's not satisfactory at all--for one thing, Crassus and Pompey had in the past always been butting heads, while Caesar and Pompey were pretty much constantly supporting each other. Pompey's appointment as sole consul (and his use of military force in the city) was due to the rioting and total civil disorder that broke out after Clodius' murder--Pompey was appointed by the senate almost without hesitation, and his movement towards the senatorial faction is well understood to have started long before 52. Already by 58 Pompey was involved in in-fighting between factions that had been allied during Caesar's consulship the year before--Clodius' threat of assassination, after Pompey publicly criticized his political methods, forced Pompey to lock himself in his house that entire year. Pompey's decision to recall Cicero from exile in 57 is usually though of as marking Pompey's real turn towards the senatorial faction, in an effort to defend himself against Clodius, who was really turning into a renegade. Certainly Cicero thought that his recall at Pompey's hands was the beginning of a much closer relationship between Pompey and the senate, although Cicero was also in total denial about the whole affair, insisting his whole life that Pompey had never actually betrayed him and coming up with absurd justifications of his belief.

Another thing is that Pompey's behavior after Clodius' death threats is somewhat odd, at least compared to his history of being a political dynamo. When Cicero returned he tried to team up with Pompey, but Pompey refused to meet with him or anyone else, retreating to his villas in the Apennines with Julia. Julia's influence on Pompey is probably pretty important. Pompey was said to be completely in love with her, and this has been offered as a possible explanation for why from about 58 until his consulship with Crassus in 55 he really didn't do a whole lot, besides recalling Cicero. Cicero comments through this period that it was pretty much impossible to see Pompey at all, that he had retired to his villas and just wouldn't come out. Certainly Pompey's political career had slowed down some--at Lucca he was the junior beneficiary, since although he and Crassus would serve as consuls Caesar's governorship of Gaul was extended and Crassus would receive the rich province of Syria. All Pompey got was to continue his governorship of Spain, which he already had control of, but he never went there or campaigned during his governorship, staying in Italy the whole time. There are some who consider Pompey's definitive break with Caesar and full-on alliance with the senatorial faction to be Julia's death in 54 (my dad is one)--certainly it was a big break, severing all need for Pompey to be nice with Caesar for family purposes alone. Importantly, Pompey married Scipio's daughter, one of Cato's henchmen. It's at this time that Crassus was killed, and it's not so clear that either Crassus' death or Julia's is more significant--I (and others) would argue that Pompey had swung toward the senate way back in 57 and that Julia's and Crassus' deaths allowed him to sever his ties with his former allies. But there are a number of explanations and theories put forward, and without the ability to actually ask Pompey what he thought it's hard to tell--the fact that nobody could get hold of Pompey for most of this period is extremely frustrating, since we have almost no insight as to what Pompey was thinking.

So there are lots and lots and lots of reasons--I've actually had to seriously compress things and have done an enormous amount of simplifying (perhaps oversimplifying). It's a very confusing and complicated period. Certainly Crassus' death had some sort of effect on relations between Caesar and Pompey (perhaps more importantly between Pompey and the senate), but it was not an isolated incident. Tension between Caesar and Pompey had been mounting almost as soon as Caesar's consulship ended, and Pompey had been increasingly allying himself with members of Cato's circle throughout that period. All of the various incidents put together led to the civil war, not a single incident in particular.

What was the effect of the news of Crassus' defeat? Enormous, to say the least. The loss of Crassus' standards in particular was of pretty great importance--legions didn't lose standards that often, certainly not to the Parthians, who really hadn't been too much of a problem before (after all, Gabinius, working for Pompey, had spanked the Parthians in 65, marching all the way into Mesopotamia, and forced them to sign a treaty with Pompey). It was a big friggin' deal, and Caesar's planned Parthian campaign in 44 was at least seen as a response to the capture of the standards. Antony's Armenian campaign was also seen the same way, although it only added to the problem because Antony let his troops starve and he ended up retreating and leaving the bulk of his army behind. Carrhae was something that the Romans had to make up for, it hung heavily in their conscious until Augustus recovered the standards--Propertius speaks of Augustus marching to the east to appease the memory of the Crassi (Crassus' elder son was killed with him, which was kind of a problem, since both Cicero and Caesar are in agreement that he was a swell guy) and their defeat.

Crassus' death left the triumvirate polarised between two men who's unwillingness to let any precede ultimately led to the downfall of the Republic. However the Late Republic was incredibly unstable and even if Pompey and Caesar never fought each other it would have collapsed eventually at the hands of another ambitious nobilis. Let's say his death sped up the death spiral of the Republic.

As for the public reaction, I think you can gauge it by the propaganda around the recovery of the army standard from Parthia (via diplomacy) under Augustus who treated it like a military victory (housing them in the shrine of his Mars Ultor temple). It was a major defeat and embarrassment to Rome.