I know less about AMD processors, but hopefully the below description of intel features also apply to AMD processors.
Intel processors are split into three ranges, in increasing performance and price - Core i3, Core i5 and Core i7. They are usually described with a 4 digit suffix, the first digit is the generation number. I think generation 8 has just come out.
In terms of performance, most articles I've read recommend the Intel Core i5 for mainstream users who care about performance, while the Intel Core i7 is made for enthusiasts and high-end users. In general, the higher the 4 digit suffix the better the performance.
Cores, i3 has 2, i5 & i7 have 4. I believe this will effect performance most, each core is effectively its own processor - the more you have the more multi-tasking the computer can do. How well software can utilise multi-core processors does vary, so by itself this may not make a program run faster.
Clock speed, measured in GHz is the number of calculations a processor can manage per a second. However, these days it is a poor indicator of performance.
Turbo Boost is available in i5 & i7, so while running at a lower clock speed (drawing less power) they can boost performance to higher clock speeds when required.
Hyper-threading allows Windows to better multi-task by treating each core as if it was 2 cores (so i3 appears as 4 virtual cores, and i7 appears to have 8 virtual cores). Again, different software utilise hyper-threading by varying amounts.
Cache - more is better, but all i3-i7 are generally okay. Cache is the processors own memory, so it can save time by looking here than moving data to and from the main system RAM.
What's difficult is knowing how all these elements will interact with your system and software, hence the general advise at the start.
If you are looking for a quicker start-up, then consider a Solid State Drive (SSD), though they tend to be more expensive. I also think you are very wise to have the two computers on the go at the same time, especially if you are looking to transfer photos with metadata such as tags. Depending on the software used, this metadata is sometimes kept in a separate file to the photos.