Discharge time of 12 V 8.4 Ah at 5 mA
I have a motorcycle battery rated as 8.4Ah (20HR). It also says 8Ah (10HR) and I am not sure what´s the difference.


The capacity decreases at higher discharge rates. The figures are telling you that discharge current of \$ \frac {8.4 \ \text {Ah}}{20 \ \text h} = 0.42 \ \text A \$ will last 20 h and \$ \frac {8.0 \ \text {Ah}}{10 \ \text h} = 0.8 \ \text A \$ will last 10 h. Given that doubling of discharge rate the total energy out is actually surprisingly close.


  8.4 from battery / 0.005 consumption = 1680 hours (70 days).


Correct method. (I didn't check your numbers but they look right.)


  Why this motorbike battery is so heavy then if it's only double the capacity? Is it because it's a 12 V?


Two reasons:


As you suspect, the 12 V is related. The energy stored in the battery is given by \$ V \times I \times t \$ so a 12 V battery will have three times the energy storage of a 4 V battery. Your 12 V battery has a capacity of \$ 12 \times 0.42 \times 20 = 100 \ \text {VAh} = 100 \ \text {Wh}\$.
Battery chemistry. Your mobile phone battery is a lithium based battery. According to Green Transportation :


Lead acid energy density is 33 - 42 Wh/kg so we would expect your battery to weigh about 3 kg. 
Lithium ion energy density is 100 to 265 Wh/kg so we could get the same energy storage in a 0.4 kg battery.





Figure 1. Energy densities for various technologies. Source: EPEC.