Link ideas: sin embargo, por eso, aunque, mientras, ya que
Sin embargo (however) sets two sentences against each other. It usually follows a full stop or semicolon and is wrapped in commas.
El piso es caro; sin embargo, lo vamos a comprar. (The flat is expensive; however, we are going to buy it.)
No me gusta el fútbol. Sin embargo, veo todos los partidos de mi hijo.
Despite appearances, it has nothing to do with embargoes — treat it as one fixed unit.
Por eso (that is why) introduces a consequence. Ya que (since, given that) introduces a cause that is already known.
Llueve; por eso no salimos. (It is raining; that is why we are not going out.)
Ya que estás aquí, ayúdame con esto. (Since you are here, help me with this.)
Careful: ya que has nothing to do with "already" here — it is a cause connector.
En cambio (on the other hand) contrasts two people or things. Mientras alone = while (time); mientras que = whereas (contrast).
Yo trabajo; en cambio, mi hermano descansa. (I work; my brother, on the other hand, rests.)
Mientras ceno, veo la tele. (While I have dinner, I watch TV.) — time
Yo ahorro, mientras que tú lo gastas todo. (I save, whereas you spend everything.) — contrast
Aunque with the normal indicative concedes a real fact: although, even though. Note that aunque already contains que — never add another.
Aunque llueve, vamos a salir. (Although it is raining, we are going to go out.)
Aunque es caro, lo voy a comprar. (Although it is expensive, I am going to buy it.)
With the subjunctive, aunque shifts from fact to hypothesis: even if. At B1, learn to recognise the difference.
Aunque llueve, salimos. (Although it IS raining — a fact.)
Aunque llueva, saldremos. (Even if it rains — we do not know yet.)
One vowel changes the meaning: llueve (fact) vs llueva (possibility).
Traps for English speakers
These are the errors English speakers make most often.