The general principle: styles come in and out of fashion (sometimes due to technical limitations changing, often just because of whimseys of fashion). When a style goes out of fashion, it becomes associated with the era when it was in fashion, making things done in that style look of that era (therefore, dated).
The halo effect of being in fashion can also mask problems and limitations with designs and styles. When the halo has gone, these flaws leap out, and the design looks dated: "Why did they do that?".
So, for example, the 'Skeuomorphic' styles vs 'Flat Design' styles shift (as linked to by Anonymous) explains the Google Chrome example: the mid/late 2000s logo follows skeuomorphic design in that it mimics real-world textures and implies real-world physical interaction, and was made in a time (2006-2011?) when that was in fashion. The new logo is an example of 'flat design' with its very very soft muted shading from a time (2010-2016?) when that style is/was in.
The old design already looks dated because it's of a style that is slipping/has slipped out of fashion. In a few years, it'll look 'so 2000s' in the same way that Cooper Black looks 'so 70s'. The same thing will probably happen to the new logo in the mid/late 2010s.
Illustration from nmmr.nl of "so 2007" skeumorphism vs what may be considered "so 2012" flat design:
I'm no design history expert but you can find a design trend for each 'dated' looking style, and/or, some practical or technological reason why that style was popular for a period then abandoned when it was no longer necessary.
For example, the Bell logo reminds me a bit of the IBM logo (1972, based on a 1960s design)...
...which Wikipedia tells me was heavily influenced by the desire to avoid the technical limitations of, first, photocopiers, then dot matrix printers: one colour, avoiding large blocks of colour, avoiding too-fine bands or details.
The old BP logo reminds me of some things which in the 20th century would have had positive associations for British motorists: for example the badge design of british carmaker Rover, the styling of iconic UK haulage firm Eddy Stobbart... The particular shade of green is close to what is sometimes called British Racing Green, harking to mid-20th century motor racing.
Once, this sort of style would have been re-assuringly classic - but with Rover and other similar companies going badly bust in the 1990s, and BP and motor racing going international, styles moved on. The look arguably went from being associated with the golden age of British motoring, to a parochial, dirty, inefficient era of highly polluting old bangers, oily petrolhead bores and doomed industries.
Then there are practical considerations. The old design would be constricted by the need to stand out very clearly in headlights regardless of dirt, grime or soot on the sign. The new BP logo looked very modern and fresh when it first came out - no other petrol stations had a logo so intricate. I'd be surprised if this wasn't influenced by some change in materials (more reflective, more resistant to soot?) that made this practical.
For any dated-looking design, there's usually some explanation like this. The types discussed above:
- A style was popular in X era until the style went too far and/or people got sick of it (e.g. skeumorphism and the old Chrome logo)
- A style had positive associations that soured (e.g. BP)
- A style was driven by a workaround to some technical limitation. Then, when new technology made new things possible, those new things looked like a breath of fresh air and everything done within the boundaries looked of the earlier era (e.g. Bell / IBM and maybe also BP).
Then there's also the flipside:
- A style was a reaction to another style on its way out. At first, this new style seemed bold, modern etc - until it became clear that there was a good reason why things weren't done like that before. After a brief flash in the pan, the style dies, forever associated with (and literally "dated" by) that brief period when it was popular.
Most fashion statements (think 1980s shell suits) fit in this 'rebound' category - "Hey, we can do this now!" followed shortly by "Why are we doing this?!?!!". An example of a rebound style we're going through now might be super-thin low contrast type on digital devices - a reaction to the possibilities offered by high-pixel-density screens, but which brings practical problems of its own.