I can easily tell them apart by their appearance and behaviour.
New money love to display very evidently expensive clothes and accessories from well recognisable luxury brands. A new shiny Rolex Submariner (or similar looking model) screams new money like nothing else (I give the benefit of the doubt, maybe, when I see it on very young people). Mine instead (for example) is vintage and has belonged to my family for half a century. Talking about jewellery, big and/or numerous diamonds are a new money fetish since they reflect their ostentatious attitude. Back to clothes, old money prevalently dress in a way that I would define classic, conservative and elegant.
Behaviour. Old money tend to be well-behaved, elegant, snobbish, classy and possibly more discreet, reserved and stiff. Typical new money behaviours are being arrogant and loud and showing off. However, behaviours are defined much more by what kind of person you are than by money, so this often doesn't apply.
Outlook. New money are sometimes clueless to the whole game of class and wealth. You might find them boosting their excessively wide, curved, 4K, 3D TV or placing too much attention on cars. This is something new money care disproportionately about (they are informed and often discuss it), their car is usually just above what they can reasonably afford, while old money ones are usually an understatement (they often even care about fuel consumption). Old money regard certain luxuries as excessive (a really huge yacht), while new money have no such notion. I could go further on boats and say, old money=sailboat, new money=motorboat/yacht, but nowadays that would be only partially accurate. Old money have a certain superiority attitude and even mild disdain for self-made people and excessive financial ambition. They would never want to trade their position with new money and never envy their assets/lifestyle because they immediately, ex ante, a priori, set them aside as nouveau riche and therefore ""inferior"". On the contrary, plain money is central for new money, who often compete with everyone else by outspending/keeping up with them. New money families place much more focus on hard work than old money.
Work and holidays. New money can't stop working (70 years old CEO, I'm looking at you, greedy ******), old money can't be bothered starting. Being overworked, new money have to roll from a holiday to another, they have a disproportionate amount of breaks in every fashionable destination (they want to be able to say, "I was in Monte Carlo/Ibiza/Maldives/St. Moritz last month".). Old money don't need actual holidays since they don't strictly work (no occupation in the traditional sense of the word), but indeed they travel a lot.
Houses. A house is a house for old money, even if it resembles a castle. For new money, it's a villa, a penthouse, a loft whatever. Some say that old money houses are cold, while new money are excessively hot. This naive distinction is due to the fact that old money houses are older and belong to family and relatives, while new money tend to have a range of new, designer furnished pied à terre in different global partying locations.
Class. While old money are necessarily upper or uppermiddle, new money might preserve or inherit their working or middle class attitude.
Spending and finances. While it is generally believed that new money spend more, I must note that this depends on how much money one has rather than how old this money is. Also, generally, old money are impoverished/impoverishing, new money, necessarily, enriching.
This post may sound offensive towards new money. I want to stress the fact that I was dealing with generalisations and stereotypes and that I know lovely, sweet and respectable new money families who only display a fraction of these behaviours. How people are isn't defined by how old their money is, but you should still be able to use this model to recognise new or old money behaviours and characteristics in people.