The architecture is the same, the frequency can be the same, the instruction sets are almost all the same (Semprons don't have SSE3 yet), cache doesn't matter in most applications, and the production process is exactly the same. Both chips come from the same factory. Semprons are just Athlon 64s that don't make the cut.
And Semprons are more than powerful enough to run a basic box from just office applications and the internet or a something like a file server. If I were to build my girlfriend a new computer it'd be a Sempron because she doesn't need anything better than that.
No,
1. The architecture can be or can not be the same. The first Semprons s462(aka AthlonXP-Throton/Barton) have different architecture.
Anyway there are s754 Semprons based on K8 without the x86-64 extensions(I think it depends on the rev, E3 & E6 do support x86-64). Also some of the low-numbered models does not support CNQ.
2. There are Semprons and Athlon64s with and without SSE3, depends on the core.
3. Cache does matter a lot in most applications, but the performance boost due to more cache is not growing linearly as more cache is added.
For multimedia(movies, mp3) and other streaming software the cache does not matter. For complex applications such as programming languages, MS Office-like apps, Graphics design apps, Movie mounting&mixing apps, the cache does matter a lot.
4. Production process is not the same. There are both Athlon64s & Semprons based on both 130nm & 90nm and SOI 1st & 2nd generation.
5. They are not produced from the same factory, there are more than one AMD fab producing chips. Also they are produced on different quality and different size wafers.
6. Every girl needs better than new PC.