Ok, so since it has been two years and no-one has answered, and this is indeed a very interesting question, here is my attempt to shed some light on it.
Is there any difference in brain activity while doing something you love or doing something similar you don't like?
Yes, without getting too deep into specifics - there certainly difference. Primarily these are that doing something that you love will produce more dopamine than doing something you dislike [e.g., 1]. Dopamine is linked to motivation to act and energy hence why it is easy to do more of the things that you enjoy .
If our emotions do affect our brain activity significantly, would it
be possible to psychologically train ourselves to like/dislike a
certain activity we want to do more/less?
Yes, I think this can be done by managing the rewards involved. Think about the dopamine production - how do you maximise this, while minimising other detrimental factors (e.g., stress/cortisol).
Here are some ways to do this:
- Break down the task so that you get a dopamine boost everytime you get one part done [e.g., 1]
- Manage your expectations - much of reward is about expectations. If you want to be better than before when doing something, as opposed to be the best, then you are much more likely to find something.
- Listen to music - in some cases this can raise dopamine levels to ensure that you have sufficient motivation to work 
- Consume substances that will increase dopamine, like st johns wort, or its precursors, like Tyrosine.
- This article  provides a set of potentially useful approaches that may be of use. However, before adopting them I would investigate the literature cited.
 Hoebel, B. G., Rada, P. V., Mark, G. P. and Pothos, E. N. (1999) 'Neural systems for reinforcement and inhibition of behavior: Relevance to eating, addiction, and depression'.
 Salimpoor, V. N., Benovoy, M., Larcher, K., Dagher, A. and Zatorre, R. J. (2011) 'Anatomically distinct dopamine release during anticipation and experience of peak emotion to music', Nat Neurosci, 14(2), 257-262.
I am happy to clarify or expand on any part of this answer if you want, so just let me know.