Any difference between Intel® i3 multi-core processor and Up to Intel® i7 multi-core processor?
Yes. (you're going to have to ask the question properly to get the answer you want, I'm not sure what you're asking.)
Secondly, I need a computer where i can log into my work remotely, any suggestions so i will have fast speed or does that just depend on my interenet provider?
Any decent computer will work for remote log-on via LogMeIn, Citrix, VPN + Remote Desktop, etc. If all you're doing is desktop sharing then the processing happens at the host (the computer you're accessing remotely), and the client (the computer you're using for remote access) doesn't really need much. An i3 will be more than enough.
If you have a decent computer (client), speeds will definitely depend on your internet connections (both at the client and at the host). The connection quality has a noticeable effect on performance/speed/responsiveness. If your host is slow, (i.e. it works slow when you sit in front of it too), it won't get any quicker by accessing it remotely...