Damaged body collection
In both the 1973 film and the TV show, the robots (film) and hosts (series) are regularly examined and collected by Westworld park staff, in order to prepare them for potential transport to the body repair departments of the park.
Body Repair Shop
In both the 1973 film and the TV show, the robots (film) and hosts (series) are repaired in hospital-like settings - the robot repair lab in the film and the Livestock Management department in the series. Employees are dressed in scrubs and perform surgery on the robots or hosts.
The Westworld park in the TV series, owned by Delos Incorporated, operates similarly to the Westworld park of the Delos resort in the film. Guests are able to do anything they want with the hosts, guns are unable to harm guests and hosts are forbidden from killing guests.
In addition, like Delos in the Westworld film, the Delos Destinations subsidiary also runs several different themed parks besides Westworld itself. (However, they differ in their choice of setting, Westworld being the only park common to both the film and the series.)
The Gunslinger on Sub-Level B82
While looking through Sub-Level B82, Bernard's flashlight briefly illuminates The Gunslinger, from the 1973 film Westworld. The figure seen in the episode is a statue, made by Nick Marra. There's a link to his (delighted) instagram post on the subject here.
In the episodes "The Original" and "The Bicameral Mind", Dr. Robert Ford mentions that "a simple handshake" would give one of the early hosts away. In the Westworld film, androids could be identified by their unrealistic looking hands.
Peter and John / William and Logan
William and Logan have some similarities to Peter Martin and John Blane respectively. (Young) William, like Peter, is a first time visitor to the park, while Logan, like John, serves as a guide to William. Peter was also recently married to a woman named Julie, whereas William is engaged to be married, to a woman named Juliet.
In addition to anachronistic covers of popular music and references to the music of Western films, Ramin Djawadi's soundtrack for the HBO TV series also includes some references to Fred Karlin's soundtrack for the original film from 1973.
The most obvious examples of this are the tracks Nitro Heist and Sweetwater, two related leitmotifs with a very similar underlying melody. Both seem to be based on two equally related tracks from the 1973 soundtrack, Chase from Westworld and Chase from Westworld - Part 2. These are heard during the famous chase sequence of the film, when The Gunslinger pursues the shocked Peter Martin on horseback and on foot through much of Westworld and the whole Delos resort. The melodies of the two TV series' tracks have a slower tempo than the tracks from the original film.
Another theme from the original film reinterpreted and adapted for the TV series is The Gunslinger. This is a sinister-sounding leitmotif for the titular character, the film's main robot antagonist. It features the use of odd sound effects and ambient elements, including artificially sounding reverbations, ripples and thuds that also make an appearance in the two Chase themes. These thudding sounds make an appearance in the series' major host theme, Freeze All Motor Functions.
Though the electronic ambient tracks Robot Repair from the film and Freeze All Motor Functions from the series do not seem to share a melody, they are otherwise similar compositions, and commonly used as the leitmotif of the robots or hosts. In a reversal to the other music reworked for the series (which is usually slower than the originals), the film's robot leitmotif is slow, mysterious and contemplative, while the series' host leitmotif has a faster tempo and a darker, more brooding quality.
Fleeing from deadly robots, from one park to the next
The final third of the original film focuses on Peter's frantic escape from Westworld, with the merciless and deadly Gunslinger hot on his heels. John is shot dead by The Gunslinger, prompting Peter to fully realise something has gone wrong with the robots' programming, and attempt to flee. Resting briefly, he stumbles upon a stranded technician who offers him some vague advice for dealing with the Gunslinger. As the pursuit continues, Peter eventually crosses the official boundaries of Westworld and enters the territory of Roman World. The Gunslinger continues to track him through Roman World, follows him through the Delos staff areas and catches up with him in Medieval World. Peter manages to narrowly defeat him and survive.
In the second season episode Virtù e Fortuna, the opening, pre-credits plot focuses on Emily during her visit to The Raj. Once the host uprising begins, Emily and Nicholas discover dead guests, prompting Emily to insist something has gone wrong. They are attacked by their host guide Ganju, who kills Nicholas. Emily shoots Ganju in self-defence and decides to run until she reaches the boundaries of Westworld. Nearing them, she briefly rests, but comes under a surprise attack by a Bengal tiger host, and resumes her escape. The tiger crosses the official border of The Raj and follows her into Westworld. Emily manages to narrowly defeat the tiger by shooting it before they fall into the water reservoir on the borders of Westworld.
Fittingly, Emily is the daughter of William, the series' counterpart to the film's Peter.
Delos' nefarious plans with their customers
In Futureworld, the 1976 sequel to the original film, the Delos company has hatched a secret scheme to create android-like clones of world leaders, important politicians and famous personalities who have visited the park. They plan to use the clones as faithful lackeys, in order to influence political and economic developments in various countries around the globe, effectivelly granting the company enormous behind-the-scenes power.
In the HBO series, Delos Incorporated having ulterior motives is first hinted at in the pilot episode of the first season, particularly in a conversation between Theresa Cullen and Lee Sizemore. In the second season, the full extent of the secret projects gradually comes to light. After his gradual takeover of Delos, William not only ensured the continued operation and development of the company's theme parks, but also orchestrated a secret data collection programme from the guests (as he detailed to James Delos during takeover negotiations) and a secret project to transfer recorded human consciousness into host bodies. The latter proves far more difficult than anticipated. The revelations about William's secret projects form a major part of the overarching story of the second season.
Setting the android creations free
The main antagonist of the short-lived Beyond Westworld series is Simon Quaid (played by James Wainwright). Quaid is retroactively identified as the creator of the robots seen in both the original film and the 1980 series. His main intention is to set his creations free and let them loose into the real world. They will help him take over and displace human beings as rulers of the planet, which in his opinion will ensure a better and more orderly world.
Quaid's character has loose parallels with that of Dr. Robert Ford (played by Anthony Hopkins) from the HBO series. Though Ford shares the intention of setting his creations free and doesn't think highly of fellow humans, he is a more morally ambiguous antagonist than Quaid. Unlike Quaid, Ford also co-developed the hosts with his colleague Arnold Weber, rather than on his own. Arnold's death proved one of the catalysts in Ford's greater interest in the hosts and his eventual conviction that they might be worthy of replacing human beings.
Colt SAA variants popular among characters
In a subversion, the series' character of William, who has intentional parallels with both Peter and The Gunslinger from the original film, prefers the Colt 1851 Navy during his first visit of Westworld. He later switches to using a custom LeMat revolver.
The Monorail of the HBO series fulfills the same role as the passenger hovercraft from the original film, ferrying visitors from the outside world to the hub area of the Delos parks, where they are greeted by staff and guided to the parks' entrances. When introduced, both William and Logan are seen seated on the right side of the vehicle, similarly to Peter and John in the original film. (In the series, it's not entirely clear whether the monorail is the sole access to the outside world, or whether it carries its passengers from an airport servicing the parks.)
The steam train of the series, used for transporting visitors from the hub area to the Westworld park, is a more elaborate replacement for the stagecoach from the original film. Upon arrival at the hub town in Westworld, a freshly disembarked female passenger expresses her awe at the sight - this occurs both in the film and the pilot episode of the series.
The park staff of the original film used three-wheeled maintenance carts in white livery. The park staff of the HBO series use all-terrain staff buggies in white livery, first seen in action during the events of the second season.