The source of my definitive "No" above comes down to this PDF from the Dell site. If you look under "Specifications", it says "Memory Type: 667-MHz, 800-MHz DDR2 SDRAM" (that's a direct quote from the setup guide).
The ThinkCentre M83 has a Haswell processor, thus its platform chipset uses DDR3. There are no Haswell chipsets that support DDR2, and I have determined that your Vostro supports DDR2 only; therefore, these two systems use incompatible RAM. QED.
Compatibility: Memory and other components
You can't put DDR3 in a DDR2 motherboard, or vice versa. There are no motherboards at the Haswell generation which support both DDR2 and DDR3; there were a few such motherboards available with hybrid DDR2/DDR3 support in earlier series of platforms, but support for DDR2 completely ended as of the Sandy Bridge era, which is a few generations older than Haswell. Read below for more details on all these code names.
As far as compatibility, basically every component in a system from the Core 2 era will have to be replaced, when moving to a Haswell system. I advise that you don't try to keep anything except the hard disks, and any USB peripherals you may have. Even the hard disks will be significantly slower, older, and lower capacity than what you will get out of the box with your new Haswell system, so you may be disappointed with the relatively lower performance of the old disks once you see what the new ones can do.
Software compatibility should be very nearly the same, assuming you can find drivers for your operating system for the new hardware. If you ran XYZ operating system on your Vostro, with a bunch of custom software, the whole software stack should run just fine (but faster) on the new system -- the only difference being, if you are trying to install a very old operating system, it may not support the much newer platform chipset (Haswell and its associated Lynx Point motherboard) without updated drivers. If you have any doubt about whether your OS will support the hardware, I advise that you run Windows 7 SP1 or newer. Windows 8.1 will also work fine.
The Intel processor series have been released in lock-step with platform chipsets (motherboards, basically) for the past half-decade or so. Very generally, the processor series and their branding has been:
"Core 2" -- a series of processors based on a major architectural shift in Intel's CPU design, which came after the "Pentium D" and "Core Duo" (note no "2"), but before the processors branded "Core i7". There were several releases with incremental improvements in the Core 2 "era", but the first chip was codenamed Conroe. Nearly all Core 2 platforms used DDR2 memory. Notably, the Core 2 era saw the end of totally custom, non-Intel platform chipsets, meaning that, all the processor generations mentioned after this one run only on a motherboard that is fundamentally an Intel part (though large and significant components on the motherboard, such as the SATA controller and network card, can still be non-Intel components to this day). This transition is owed in part to the fact that Core 2 was the last CPU to use a memory controller on the motherboard; the newer CPUs moved the memory controller onto the CPU package (if not in the CPU die itself).
"Core i7", first generation -- this marks the first processors whose system boards supported the improved DDR3 memory (which is faster in many ways). The number on the end of the "DDR" just refers to how advanced (and new) the memory "era" is. DDR2 was mainstream for about 4-5 years before DDR3 took over. Looking ahead, it's looking like DDR4 will enter the mainstream in the 2015-2016 timeframe, or 2017 at the latest. These first-generation Core i7 processors were codenamed Nehalem.
"Core i" (i3, i5, i7), second generation -- this marks the second generation of processors with the "Core i7" branding, though they also introduced "i3" and "i5" as cut-down, cheaper versions. These also support DDR3, and came with a new motherboard chipset. They were codenamed "Sandy Bridge", and are still considered new enough to be useful for mid-range to high-end workloads even today, although the newest processors can be substantially faster.
"Core i" (i3, i5, i7), third generation -- this marks the third generation of processors with the "Core i7" branding, though they also have "i3" and "i5" models as cut-down, cheaper versions. These also support DDR3, and the CPUs can be installed into the previous generation's motherboards, too. Not a whole lot here changed, except for a dramatic improvement in on-board graphics performance, and a dramatic improvement in power efficiency (the amount of performance you get per watt of power invested). They were codenamed "Ivy Bridge", and are considered fast and modern enough for most workloads in Q3 2014.
"Core i" (i3, i5, i7), fourth generation -- as of July 2014, these are the latest released Intel mainstream processors. They have the same branding as the previous generation, but are referred to as "Fourth-generation Intel Core processors", or similar. They are slightly faster and more power-efficient than their third-generation close cousins, but they still support DDR3 (only). They are codenamed "Haswell". Intel moved away from the "Bridge" codename because Haswell represents a new CPU microarchitecture, meaning that Haswell introduced significant new CPU instructions. Still, for ordinary workloads, Haswell isn't much faster than Ivy Bridge.
When overviewing the performance of these generations and the features offered, the big bullet points are thus:
Conroe introduced advanced vector processing instructions called SSE3, which makes Intel CPUs better at processing workloads which afford massive parallelization. It also vastly improved virtualization performance.
Nehalem introduced DDR3 support and PCI-Express 2.0, which enormously improved memory and graphics performance, respectively. This can make the difference between a slow, unusable system and a fast, responsive one. Nehalem also got rid of the Front-Side Bus (FSB), a venerable but inefficient bottleneck in the old system design, replacing it with point-to-point connections between the CPU and the PCI Express lanes, called Quick Path Interconnect.
Sandy Bridge introduced the DMI 2.0 interface, replacing QPI in mainstream desktop processors (we barely knew ye!). This further improved the bandwidth and latency between core system components such as the memory, PCI Express lanes, networking chipsets, and the CPU. Sandy Bridge also upped the PCI Express support to 3.0, roughly doubling the available bandwidth for add-on cards such as GPUs and network cards.
Ivy Bridge introduced USB 3.0 support built into the platform (there were some Sandy Bridge motherboards supporting USB 3.0, but it was not a "native" feature of the CPU/platform). Ivy Bridge also vastly improved graphics performance on Intel CPUs that have a GPU, and power efficiency, especially on mobile (laptop) devices.
Haswell further refined the trends taking place in Ivy Bridge's development by improving GPU performance and power efficiency, while also giving encryption performance a boost, and adding some features demanded by enterprise customers, such as TSX. Haswell also improved overall system performance by a modest amount (the other releases did, too, but Haswell's and Ivy Bridge's were both rather modest, especially considering neither platform has updated the link between the CPU and the peripherals, nor the memory throughput).
You will be upgrading from Allendale all the way to Haswell, which is an enormous leap. All the features are cumulative, so you'll be getting USB 3.0, PCI-Express 3.0, DDR3, DMI 2.0, the whole shebang. Should be quite an upgrade for you!